1. Write the function in standard form y=ax^2+bx+c.
2. Find the vertex by using the formula x=-b/2a.
3. Find the axis of symmetry by setting x=-b/2a.
4. Sketch the parabola using the vertex and axis of symmetry as guides.
4.1 quadratic functions and transformationsleblance
This document discusses quadratic functions and their transformations. It defines key terms like parabola, vertex, and axis of symmetry. The vertex form of a quadratic function makes it easy to identify transformations - the a value determines stretching or compression, the h value shifts the graph horizontally, and the k value shifts it vertically. For any quadratic function, the minimum or maximum value will occur at the vertex. Graphing involves plotting the vertex and axis of symmetry, then using a table of values. Writing quadratic functions starts with identifying the vertex coordinates, then using another point to solve for the a value.
This document discusses solving quadratic equations by graphing related quadratic functions. It begins with examples of graphing quadratic functions of the form y=ax^2+bx+c and finding the x-intercepts, which are the solutions to the corresponding quadratic equation. Next, it provides examples of applying this process to solve application problems involving projectile motion. It concludes with a lesson quiz to assess understanding.
210 graphs of factorable rational functionsmath260
The document discusses vertical asymptotes of rational functions. It provides examples of the functions y=1/x and y=1/x^2. For y=1/x, the graph does not touch the vertical asymptote at x=0, but gets infinitely close to it as x approaches 0. The graph runs upward along the right of the asymptote and downward along the left. For y=1/x^2, the graph also has a vertical asymptote at x=0 and runs upward along both sides of the asymptote. Vertical asymptotes occur where the denominator is 0.
The document discusses graphing quadratic functions. It defines a quadratic function as an equation of the form y=ax^2 +bx+c where a ≠ 0, forming a U-shaped parabola. It explains that the vertex is the highest/lowest point and the axis of symmetry is the vertical line through the vertex. It provides steps for graphing a quadratic function by finding the vertex coordinates using the standard form equation, choosing x-values on each side of the vertex, calculating corresponding y-values, and plotting points connected with a smooth curve. An example problem demonstrates these steps.
LESSON-Effects of changing a,h and k in the Graph of Quadratic FunctionRia Micor
The document discusses transforming quadratic functions into vertex form and describes how changing the values of a, h, and k affects the graph of the function. It then has students work in groups to graph and describe quadratic functions based on given equations in order to understand how the vertex, opening direction, and any shifts in the vertex position are represented algebraically.
Graphing Quadratic Functions in Standard Formcmorgancavo
This document discusses graphing quadratic functions of the form y = ax^2 + bx + c. It provides the following key points:
- Quadratic functions produce parabolic graphs that open up or down depending on whether a is positive or negative.
- The vertex of the parabola is the point of minimum or maximum, which corresponds to the line of symmetry that passes through it.
- To graph a quadratic, one finds the line of symmetry, determines the vertex coordinates, and plots at least four other points to connect into a smooth curve.
Math 4 lecture on Graphing Rational FunctionsLeo Crisologo
The document discusses graphing rational functions. It reviews key concepts such as defining rational functions as the ratio of two polynomial functions, and determining their domain. It also covers identifying horizontal asymptotes based on the degrees of the numerator and denominator polynomials. The document provides examples of finding the horizontal asymptote for different rational functions and summarizes the different cases for the horizontal asymptote based on the degree relationship between the numerator and denominator.
Quadratic functions are modeled by the equation y = ax^2 + bx + c, where a ≠ 0. They produce U-shaped parabolic graphs. Many real-world phenomena follow quadratic patterns, like water in a fountain or a basketball's trajectory. The lowest or highest point on a parabola is the vertex. To graph a quadratic function in standard form, you first find the vertex by calculating -b/2a, then plot the axis of symmetry and other points to sketch the parabolic curve.
4.1 quadratic functions and transformationsleblance
This document discusses quadratic functions and their transformations. It defines key terms like parabola, vertex, and axis of symmetry. The vertex form of a quadratic function makes it easy to identify transformations - the a value determines stretching or compression, the h value shifts the graph horizontally, and the k value shifts it vertically. For any quadratic function, the minimum or maximum value will occur at the vertex. Graphing involves plotting the vertex and axis of symmetry, then using a table of values. Writing quadratic functions starts with identifying the vertex coordinates, then using another point to solve for the a value.
This document discusses solving quadratic equations by graphing related quadratic functions. It begins with examples of graphing quadratic functions of the form y=ax^2+bx+c and finding the x-intercepts, which are the solutions to the corresponding quadratic equation. Next, it provides examples of applying this process to solve application problems involving projectile motion. It concludes with a lesson quiz to assess understanding.
210 graphs of factorable rational functionsmath260
The document discusses vertical asymptotes of rational functions. It provides examples of the functions y=1/x and y=1/x^2. For y=1/x, the graph does not touch the vertical asymptote at x=0, but gets infinitely close to it as x approaches 0. The graph runs upward along the right of the asymptote and downward along the left. For y=1/x^2, the graph also has a vertical asymptote at x=0 and runs upward along both sides of the asymptote. Vertical asymptotes occur where the denominator is 0.
The document discusses graphing quadratic functions. It defines a quadratic function as an equation of the form y=ax^2 +bx+c where a ≠ 0, forming a U-shaped parabola. It explains that the vertex is the highest/lowest point and the axis of symmetry is the vertical line through the vertex. It provides steps for graphing a quadratic function by finding the vertex coordinates using the standard form equation, choosing x-values on each side of the vertex, calculating corresponding y-values, and plotting points connected with a smooth curve. An example problem demonstrates these steps.
LESSON-Effects of changing a,h and k in the Graph of Quadratic FunctionRia Micor
The document discusses transforming quadratic functions into vertex form and describes how changing the values of a, h, and k affects the graph of the function. It then has students work in groups to graph and describe quadratic functions based on given equations in order to understand how the vertex, opening direction, and any shifts in the vertex position are represented algebraically.
Graphing Quadratic Functions in Standard Formcmorgancavo
This document discusses graphing quadratic functions of the form y = ax^2 + bx + c. It provides the following key points:
- Quadratic functions produce parabolic graphs that open up or down depending on whether a is positive or negative.
- The vertex of the parabola is the point of minimum or maximum, which corresponds to the line of symmetry that passes through it.
- To graph a quadratic, one finds the line of symmetry, determines the vertex coordinates, and plots at least four other points to connect into a smooth curve.
Math 4 lecture on Graphing Rational FunctionsLeo Crisologo
The document discusses graphing rational functions. It reviews key concepts such as defining rational functions as the ratio of two polynomial functions, and determining their domain. It also covers identifying horizontal asymptotes based on the degrees of the numerator and denominator polynomials. The document provides examples of finding the horizontal asymptote for different rational functions and summarizes the different cases for the horizontal asymptote based on the degree relationship between the numerator and denominator.
Quadratic functions are modeled by the equation y = ax^2 + bx + c, where a ≠ 0. They produce U-shaped parabolic graphs. Many real-world phenomena follow quadratic patterns, like water in a fountain or a basketball's trajectory. The lowest or highest point on a parabola is the vertex. To graph a quadratic function in standard form, you first find the vertex by calculating -b/2a, then plot the axis of symmetry and other points to sketch the parabolic curve.
The document discusses different types of functions including:
- Constant functions which always take the same output value.
- Linear functions which are polynomials of degree 1 that pass through the origin.
- Quadratic functions which are of the form y = ax2 + bx + c and graph as a parabola.
- Rational functions which are the quotient of two polynomials.
- Absolute value functions which output the absolute value of the input.
It provides examples of how to determine the domain and range of each type of function by analyzing their graphs or algebraic expressions. Key aspects like intercepts, vertex, and asymptotes are also examined.
This document discusses modeling with quadratic functions. It provides examples of writing quadratic functions in vertex form, intercept form, and standard form given certain information about the parabola. It also discusses finding a quadratic model from a data set and using it to find the maximum point.
This document discusses graphing quadratic functions. It defines a quadratic function as having the form y = ax^2 + bx + c, where a is not equal to 0. The graph of a quadratic function is a U-shaped parabola. It discusses finding the vertex and axis of symmetry in standard form, vertex form, and intercept form. Examples are provided for graphing quadratic functions written in these three forms.
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
The document discusses graphing quadratic functions in standard form (y=ax^2 + bx + c). It explains that the graph is a parabola that can open up or down depending on whether a is positive or negative. The line of symmetry for the parabola passes through the vertex and is given by the equation x=-b/2a. The steps to graph are: 1) find the line of symmetry, 2) plug the x-value into the original equation to find the vertex, 3) find two other points and reflect them across the line of symmetry.
Alg II Unit 4-1 Quadratic Functions and Transformationsjtentinger
The document provides an overview of quadratic functions and their transformations. It defines key concepts like parabolas, quadratic functions, vertex form, and the parent function. It explains how to graph quadratic functions and how their graphs are transformed through reflection, stretching, compression, and translation based on changes to the coefficients in the function. Examples are provided to demonstrate finding features of quadratic functions like the vertex, axis of symmetry, minimum/maximum values, and describing the transformations.
The document discusses graphing quadratic functions. It provides an example of graphing the function y = x^2 - 2x - 9. The key steps are:
1) Assign values to x ranging from -4 to 5 and calculate the corresponding y-values.
2) Plot the points (x, y) on a graph.
3) The graph will be a parabola, which can open upward, downward, rightward, or leftward depending on the equation.
Higher Maths 1.2.2 - Graphs and Transformationstimschmitz
The document discusses different types of transformations that can be applied to functions and their graphs:
1) Translations slide the graph horizontally or vertically by adding or subtracting a value from the x- or y-coordinates.
2) Reflections flip the graph across the x- or y-axis by changing the sign of the x- or y-coordinates.
3) Distortions change the scale of the graph by multiplying all coordinates by a constant value, stretching or compressing the graph vertically or horizontally.
4) Composite transformations apply multiple transformations sequentially, such as reflecting and translating a parent graph to sketch a new transformed graph.
This document discusses graphing rational functions. It defines key concepts like domain, range, intercepts, zeros, and asymptotes. An example rational function f(x)=x-2/(x+2) is used to demonstrate how to find these values and graph the function. The domain is all real numbers except -2, the x-intercept is 2, and the y-intercept is -1. The vertical asymptote is at x=-2. Horizontal asymptotes occur when the degree of the numerator is less than, equal to, or greater than the degree of the denominator.
This document defines key concepts about linear equations including slope, graphing lines, finding the slope between two points, and writing equations of lines in various forms (point-slope, slope-intercept). It provides examples of finding the slope and equation of a line given points, graphing lines, and identifying vertical and horizontal lines. The objectives are to define slope as a rate of change, graph a line given a point and slope, find the slope from two points, and write equations of vertical and horizontal lines in different forms. Practice problems are included to reinforce these concepts.
This document provides 28 examples of graphs of different types of rational functions. The rational functions shown include those of the form y=1/x, y=-1/x, y=(x+a)/(x+b), and others with variations in the numerator and denominator polynomials. Each example graphically depicts a different rational function to illustrate their key characteristics and behaviors.
This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.
Alg II Unit 4-2 Standard Form of a Quadratic Functionjtentinger
This document discusses the standard form of a quadratic function y = ax^2 + bx + c and provides examples of:
- Converting quadratic equations between standard and vertex form
- Using the properties of quadratic functions to find the vertex, axis of symmetry, minimum/maximum values, y-intercept
- Graphing quadratic functions by hand and with a calculator to identify key features
It also provides examples of interpreting real world applications that can be modeled with quadratic functions.
This document discusses graphing equations. It begins by stating the objectives of graphing equations, which are to draw graphs of equations, define and find intercepts on the axes, study symmetry with respect to the axes and origin, and determine properties of equations. It then provides examples of constructing graphs of equations by selecting x-values, evaluating the equation, plotting the points, and connecting them. It also discusses finding the intercepts on the axes by setting y=0 and x=0 in an equation and solving for x and y, respectively. Finally, it examines symmetry of graphs with respect to the x-axis, y-axis, and origin.
The document provides examples and steps for graphing functions of the form y=ax, y=ax+k, and identifying their domains and ranges. It discusses vertical stretches, shrinks, translations, and reflections of the graph of y=x. Example 1 graphs y=3x, which is a vertical stretch of y=x by a factor of 3. Example 2 graphs y=-0.5x, a vertical shrink and reflection of y=x. Example 3 graphs y=x+2, a vertical translation of y=x up by 2 units. The guided practice graphs additional functions and identifies their transformations compared to y=x.
This document contains information about a math class that is reviewing quadratic functions. It includes:
1. An outline of the class agenda which focuses on reviewing key concepts like how the b-value affects the parabola and completing classwork.
2. Details about grading which includes assignments, homework, tests, the final exam, and notebook checks.
3. Sample problems and class notes focused on quadratic functions, including the axis of symmetry, vertex, graphing techniques, and how changing a, b, and c values impacts the parabola.
4. Examples of completing the steps to graph quadratic functions like plotting points and reflecting over the axis of symmetry.
The document discusses transformations of quadratic functions, including horizontal and vertical translations, reflections, and stretches or compressions. Horizontal translations move the graph right or left, depending on the value of h in the function f(x) = (x - h)2. Vertical translations move the graph up or down depending on the value of k. The vertex of the parabola after any transformation is located at the point (h, k). Reflections occur when the value of a in the function f(x) = a(x)2 is negative, causing the graph to reflect over the x-axis. Stretches and compressions occur when the absolute value of a is greater or less than 1, respectively.
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
The document discusses quadratic functions and equations. It defines a quadratic function as a polynomial of the second degree of the form f(x)=ax^2 + bx + c, where a ≠ 0. It describes how the discriminant determines the number of solutions and the relationship between the sign of coefficients a and d and the graph of the quadratic function. Examples are also provided to illustrate key concepts like finding the vertex, zeroes, and maximum/minimum values of quadratic functions.
This document provides instruction on graphing quadratic functions. It defines key terms like parabola, vertex, axis of symmetry, and solutions. It explains that the vertex is the highest or lowest point and its x-coordinate is the axis of symmetry. The solutions are the x-values where the parabola crosses the x-axis. Examples show how to identify the vertex and whether the function has a maximum or minimum. The last example demonstrates graphing a quadratic function using a table of values.
A quadratic function is a function with x^2 in its general form of y=ax^2 + bx + c. The vertex is the highest or lowest point of the parabola, located using the x-coordinate of -b/2a. The axis of symmetry is the vertical line passing through the vertex. Students work in groups of three, with designated roles of grapher, recorder, and summarizer to graph and analyze sets of quadratic functions based on their vertices, axes of symmetry, and coefficients.
The document discusses different types of functions including:
- Constant functions which always take the same output value.
- Linear functions which are polynomials of degree 1 that pass through the origin.
- Quadratic functions which are of the form y = ax2 + bx + c and graph as a parabola.
- Rational functions which are the quotient of two polynomials.
- Absolute value functions which output the absolute value of the input.
It provides examples of how to determine the domain and range of each type of function by analyzing their graphs or algebraic expressions. Key aspects like intercepts, vertex, and asymptotes are also examined.
This document discusses modeling with quadratic functions. It provides examples of writing quadratic functions in vertex form, intercept form, and standard form given certain information about the parabola. It also discusses finding a quadratic model from a data set and using it to find the maximum point.
This document discusses graphing quadratic functions. It defines a quadratic function as having the form y = ax^2 + bx + c, where a is not equal to 0. The graph of a quadratic function is a U-shaped parabola. It discusses finding the vertex and axis of symmetry in standard form, vertex form, and intercept form. Examples are provided for graphing quadratic functions written in these three forms.
This powerpoint presentation discusses or talks about the topic or lesson Functions. It also discusses and explains the rules, steps and examples of Quadratic Functions.
The document discusses graphing quadratic functions in standard form (y=ax^2 + bx + c). It explains that the graph is a parabola that can open up or down depending on whether a is positive or negative. The line of symmetry for the parabola passes through the vertex and is given by the equation x=-b/2a. The steps to graph are: 1) find the line of symmetry, 2) plug the x-value into the original equation to find the vertex, 3) find two other points and reflect them across the line of symmetry.
Alg II Unit 4-1 Quadratic Functions and Transformationsjtentinger
The document provides an overview of quadratic functions and their transformations. It defines key concepts like parabolas, quadratic functions, vertex form, and the parent function. It explains how to graph quadratic functions and how their graphs are transformed through reflection, stretching, compression, and translation based on changes to the coefficients in the function. Examples are provided to demonstrate finding features of quadratic functions like the vertex, axis of symmetry, minimum/maximum values, and describing the transformations.
The document discusses graphing quadratic functions. It provides an example of graphing the function y = x^2 - 2x - 9. The key steps are:
1) Assign values to x ranging from -4 to 5 and calculate the corresponding y-values.
2) Plot the points (x, y) on a graph.
3) The graph will be a parabola, which can open upward, downward, rightward, or leftward depending on the equation.
Higher Maths 1.2.2 - Graphs and Transformationstimschmitz
The document discusses different types of transformations that can be applied to functions and their graphs:
1) Translations slide the graph horizontally or vertically by adding or subtracting a value from the x- or y-coordinates.
2) Reflections flip the graph across the x- or y-axis by changing the sign of the x- or y-coordinates.
3) Distortions change the scale of the graph by multiplying all coordinates by a constant value, stretching or compressing the graph vertically or horizontally.
4) Composite transformations apply multiple transformations sequentially, such as reflecting and translating a parent graph to sketch a new transformed graph.
This document discusses graphing rational functions. It defines key concepts like domain, range, intercepts, zeros, and asymptotes. An example rational function f(x)=x-2/(x+2) is used to demonstrate how to find these values and graph the function. The domain is all real numbers except -2, the x-intercept is 2, and the y-intercept is -1. The vertical asymptote is at x=-2. Horizontal asymptotes occur when the degree of the numerator is less than, equal to, or greater than the degree of the denominator.
This document defines key concepts about linear equations including slope, graphing lines, finding the slope between two points, and writing equations of lines in various forms (point-slope, slope-intercept). It provides examples of finding the slope and equation of a line given points, graphing lines, and identifying vertical and horizontal lines. The objectives are to define slope as a rate of change, graph a line given a point and slope, find the slope from two points, and write equations of vertical and horizontal lines in different forms. Practice problems are included to reinforce these concepts.
This document provides 28 examples of graphs of different types of rational functions. The rational functions shown include those of the form y=1/x, y=-1/x, y=(x+a)/(x+b), and others with variations in the numerator and denominator polynomials. Each example graphically depicts a different rational function to illustrate their key characteristics and behaviors.
This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.
Alg II Unit 4-2 Standard Form of a Quadratic Functionjtentinger
This document discusses the standard form of a quadratic function y = ax^2 + bx + c and provides examples of:
- Converting quadratic equations between standard and vertex form
- Using the properties of quadratic functions to find the vertex, axis of symmetry, minimum/maximum values, y-intercept
- Graphing quadratic functions by hand and with a calculator to identify key features
It also provides examples of interpreting real world applications that can be modeled with quadratic functions.
This document discusses graphing equations. It begins by stating the objectives of graphing equations, which are to draw graphs of equations, define and find intercepts on the axes, study symmetry with respect to the axes and origin, and determine properties of equations. It then provides examples of constructing graphs of equations by selecting x-values, evaluating the equation, plotting the points, and connecting them. It also discusses finding the intercepts on the axes by setting y=0 and x=0 in an equation and solving for x and y, respectively. Finally, it examines symmetry of graphs with respect to the x-axis, y-axis, and origin.
The document provides examples and steps for graphing functions of the form y=ax, y=ax+k, and identifying their domains and ranges. It discusses vertical stretches, shrinks, translations, and reflections of the graph of y=x. Example 1 graphs y=3x, which is a vertical stretch of y=x by a factor of 3. Example 2 graphs y=-0.5x, a vertical shrink and reflection of y=x. Example 3 graphs y=x+2, a vertical translation of y=x up by 2 units. The guided practice graphs additional functions and identifies their transformations compared to y=x.
This document contains information about a math class that is reviewing quadratic functions. It includes:
1. An outline of the class agenda which focuses on reviewing key concepts like how the b-value affects the parabola and completing classwork.
2. Details about grading which includes assignments, homework, tests, the final exam, and notebook checks.
3. Sample problems and class notes focused on quadratic functions, including the axis of symmetry, vertex, graphing techniques, and how changing a, b, and c values impacts the parabola.
4. Examples of completing the steps to graph quadratic functions like plotting points and reflecting over the axis of symmetry.
The document discusses transformations of quadratic functions, including horizontal and vertical translations, reflections, and stretches or compressions. Horizontal translations move the graph right or left, depending on the value of h in the function f(x) = (x - h)2. Vertical translations move the graph up or down depending on the value of k. The vertex of the parabola after any transformation is located at the point (h, k). Reflections occur when the value of a in the function f(x) = a(x)2 is negative, causing the graph to reflect over the x-axis. Stretches and compressions occur when the absolute value of a is greater or less than 1, respectively.
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
The document discusses quadratic functions and equations. It defines a quadratic function as a polynomial of the second degree of the form f(x)=ax^2 + bx + c, where a ≠ 0. It describes how the discriminant determines the number of solutions and the relationship between the sign of coefficients a and d and the graph of the quadratic function. Examples are also provided to illustrate key concepts like finding the vertex, zeroes, and maximum/minimum values of quadratic functions.
This document provides instruction on graphing quadratic functions. It defines key terms like parabola, vertex, axis of symmetry, and solutions. It explains that the vertex is the highest or lowest point and its x-coordinate is the axis of symmetry. The solutions are the x-values where the parabola crosses the x-axis. Examples show how to identify the vertex and whether the function has a maximum or minimum. The last example demonstrates graphing a quadratic function using a table of values.
A quadratic function is a function with x^2 in its general form of y=ax^2 + bx + c. The vertex is the highest or lowest point of the parabola, located using the x-coordinate of -b/2a. The axis of symmetry is the vertical line passing through the vertex. Students work in groups of three, with designated roles of grapher, recorder, and summarizer to graph and analyze sets of quadratic functions based on their vertices, axes of symmetry, and coefficients.
This document provides information on applying the vertex formula to problems involving projectile motion. It discusses key concepts like projectiles, parabolas, maximum and minimum heights. Examples are provided for using the vertex formula (-b/2a) to calculate the time a projectile reaches its highest point and the maximum height achieved for scenarios like firework displays and baseballs thrown vertically.
The document discusses quadratic functions and their graphs. It defines quadratic functions as functions of the form f(x)=ax^2+bx+c, where a is not equal to 0. The graph of a quadratic function is a parabola with certain characteristics: it is symmetrical about an axis of symmetry and has a vertex which is either a maximum or minimum point. The axis of symmetry is the line x=0 for functions of the form f(x)=ax^2 and the vertex is at (0,0). For functions of the form f(x)=ax^2+k, the graph is a translation of f(x)=ax^2, so the vertex is (0,k) and the
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
- The document discusses graphs of linear and quadratic equations.
- Linear equations produce straight line graphs, while quadratic equations produce curved graphs called parabolas.
- To graph a parabola, one finds the vertex using the vertex formula, then makes a table of x and y values centered around the vertex to plot the points symmetrically.
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
02.21.2020 Algebra I Quadraic Functions.pptjannelewlawas
The document provides an agenda and lesson plan for an Algebra I class that includes the following:
1. The class will begin with a D.E.A.R activity called "Jaguar Ascensions" followed by taking Cornell notes on the lesson.
2. Students will then summarize their notes and have a teacher dialogue.
3. A number sense routine involving finding a missing number in an equation will be completed.
4. The remainder of the class will cover topics on quadratic functions including graphing quadratic functions in standard, vertex, and intercept forms as well as solving quadratic equations through factoring and using the quadratic formula.
The document discusses key concepts about quadratic functions including:
1) Quadratic functions can be represented by the equation f(x) = ax^2 + bx + c and form a parabolic shape that can open up or down.
2) The axis of symmetry of a parabola passes through the vertex (highest/lowest point) and can be found using the equation x = -b/2a.
3) The vertex coordinates can be determined by substituting the axis of symmetry value into the original equation.
4) Quadratic functions can have 0, 1, or 2 x-intercepts depending on the number of real solutions to the related quadratic equation.
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of such a function is a parabola, and discusses key features of parabolas including whether they open up or down based on the sign of a, their line of symmetry, and how to find the vertex. The document gives step-by-step instructions for graphing a quadratic function in standard form, including finding the line of symmetry, locating the vertex, and using reflection across the line of symmetry to graph the full parabola.
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of a quadratic function is a parabola that can open up or down, with the lowest or highest point being the vertex. It describes how to find the line of symmetry, which is x = -b/2a and always passes through the vertex. The steps to graph a quadratic function are given as finding the line of symmetry, plugging it into the original equation to find the vertex coordinates, then finding two other points and reflecting them across the line of symmetry.
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of such a function is a parabola, and discusses how to find the vertex and line of symmetry. The line of symmetry is given by x = -b/2a, and the vertex is found by plugging the x-value from the line of symmetry into the original equation to get the y-value. An example of graphing the function y = 2x^2 - 4x - 1 is presented to demonstrate these concepts.
This document discusses absolute value functions and their graphs. It defines the parent graph of an absolute value function as f(x)=x and describes its V-shape and axis of symmetry. Examples are given of absolute value functions after various transformations from the parent graph, along with the general form y=a(x-h)+k. The document also describes how to identify the vertex and axis of symmetry from the equation of an absolute value function and how to write the equation of an absolute value function given its graph. Homework problems are assigned from the textbook.
Finding the opening of the parabola, vertex, axis of symmetry, y-intercept, x- intercept, domain, range, and the minimum/maximum value including the illustration of the graph
1) To graph a quadratic function in standard form y=ax^2 + bx + c, one finds the axis of symmetry using the formula x=-b/2a, finds the vertex by substituting the axis into the function, and finds two other points to reflect across the axis and connect with a smooth curve.
2) The axis of symmetry always passes through the vertex. The y-intercept can be found by substituting x=0 into the function.
3) A quadratic function has either 0, 1, or 2 solutions which are found by setting the function equal to 0 and solving the resulting quadratic equation.
The document provides information about graphing quadratic functions in standard form (y=ax^2 + bx + c). It discusses that the graph is a parabola that opens up or down depending on whether a is positive or negative. It also explains that the line of symmetry for the parabola is given by x=-b/2a and the vertex is found by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a quadratic function in standard form using three steps: finding the line of symmetry, finding the vertex, and finding/reflecting two other points to connect with a smooth curve.
The document discusses graphing quadratic functions. It begins with reviewing key concepts like the vertex and axis of symmetry. The effects of the a, b, and c coefficients on the parabola are explained. Examples are provided to show how changing these values affects the width, direction opened, and vertical translation of the graph. The class will graph various quadratic functions by finding the axis of symmetry, vertex, y-intercept, and other points to plot the parabola. Students are assigned class work problems to graph quadratic functions and show their work.
This document contains instructions and examples for graphing quadratic functions. It begins with a review of key characteristics of quadratics such as their standard form and parabolic shape. The document then provides examples of graphing different types of quadratics by making tables of x-y values and plotting the points. It demonstrates how to find the axis of symmetry and vertex of a parabola by using the standard form equation. Students are instructed to graph various quadratics as class work.
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form - and how to graph each. It provides the key characteristics and steps to graph a quadratic function for each form. Standard form is ax2 + bx + c, vertex form is a(x-h)2 + k, and intercept form is a(x-p)(x-q). The document explains how to identify the vertex, axis of symmetry, intercepts, and whether the graph opens up or down based on the coefficients for each form. It also gives the process to follow to plot points and sketch the parabolic graph.
6.6 analyzing graphs of quadratic functionsJessica Garcia
This document discusses analyzing and graphing quadratic functions. It defines key terms like vertex, axis of symmetry, and vertex form. It explains that the graph of y=ax^2 is a parabola, and how the value of a affects whether the parabola opens up or down. It also describes how to graph quadratic functions in vertex form by plotting the vertex and axis of symmetry, and using symmetry.
The document discusses how to graph quadratic functions of the form y = ax^2 + bx + c. It explains that the graph is a parabola that may open up or down. It describes how to find the line of symmetry, which is -b/2a, and how this line passes through the vertex. It provides steps to find the vertex by plugging the x-value from the line of symmetry into the original equation. Finally, it demonstrates graphing a parabola by finding two additional points and reflecting them across the line of symmetry.
4.1 quadratic functions and transformationsleblance
This document discusses quadratic functions and transformations. It defines key terms like parabola, vertex, and axis of symmetry. It explains how the a value in the vertex form y=a(x-h)^2+k determines if a parabola is vertically stretched or compressed. It also states that if a is negative, the graph is reflected over the x-axis. The minimum or maximum value of a quadratic is always the y-coordinate of the vertex. The document provides examples of graphing and writing quadratic functions using vertex form.
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the definitions and formulas for each form. It then explains how to graph each form by identifying key features of the equation, finding important points like the vertex, axis of symmetry, intercepts, and connecting points to sketch the parabolic curve. Graphing techniques include using the value of a to determine the opening direction, using b and c for standard form, using h and k for vertex form, and using p and q for intercepts form.
The document discusses graphing quadratic functions. It begins with reviewing key concepts like the vertex and axis of symmetry and how the a, b, and c coefficients affect the graph. Examples are provided for determining the width, direction opened, and vertical shift based on these coefficients. The remainder of the document provides step-by-step examples of graphing quadratic functions by finding the axis of symmetry, vertex, y-intercept, and other points to plot the parabolic curve.
This document discusses graphing quadratic functions in vertex and intercept forms. It defines key terms like vertex, axis of symmetry, and parabola. It explains how to graph quadratic functions written in vertex form and intercept form. Examples are provided for graphing functions in each form. The document also discusses how to change between the vertex, intercept, and standard forms of a quadratic equation.
The lymphatic system helps destroy microorganisms, absorbs tissue fluid and transports it back to the bloodstream, and helps fight illnesses and infections. It is composed of lymph vessels, lymph nodes, lymphocytes like B and T cells, the spleen, thymus gland, tonsils, and bone marrow. Together these parts work to filter lymph, produce white blood cells, and defend the body against pathogens.
Blood transports oxygen, nutrients, hormones, carbon dioxide, and waste throughout the body. It also fights infections through white blood cells and helps regulate temperature. Blood is made up of plasma, platelets, red blood cells, and white blood cells. It exists in different blood types (A, B, AB, O) and Rh factors (+ or -) to prevent incompatible mixing. Diseases can affect blood cells like leukemia and anemia.
The circulatory system transports blood throughout the body via blood vessels. The heart pumps blood through two circuits - systemic circulation carries blood to the body and pulmonary circulation carries blood to and from the lungs. Blood flows from the heart through arteries, then narrows into smaller arterioles and capillaries where nutrients and gases are exchanged with body tissues before returning to the heart through veins. Maintaining healthy blood pressure can prevent circulatory diseases like heart attacks and strokes.
The respiratory system takes in oxygen and removes carbon dioxide through breathing. Breathing is controlled by the diaphragm muscle contracting and relaxing. Air enters the nose and is warmed and filtered before reaching the lungs. In the lungs, bronchioles branch into alveoli where gas exchange occurs and oxygen enters the blood while carbon dioxide leaves. The respiratory and circulatory systems work together to maintain homeostasis in the body.
The excretory system collects and eliminates waste from the body through various organs including the kidneys, ureters, bladder and urethra. The kidneys filter waste from the blood to produce urine, which is stored in the bladder and then passed out of the body through the urethra. The excretory system works to maintain homeostasis by regulating fluid levels and removing toxins.
The digestive system breaks down food into small molecules that can be absorbed and used by the body. Food goes through four steps - ingestion, digestion, absorption, and elimination. Digestion involves both mechanical and chemical breakdown of food. Enzymes produced throughout the digestive system aid in chemical digestion. The major organs of the digestive system include the mouth, esophagus, stomach, small intestine, large intestine, liver, pancreas, and gallbladder.
The skin is the largest organ of the body and acts as a protective barrier. It has three layers - the epidermis, dermis and fatty layer. The skin protects the body from damage, regulates temperature and moisture, produces vitamin D, and detects sensations like touch, temperature and pain. When injured, the skin repairs through processes like scabbing, bruising and wound healing. The skin works to maintain homeostasis by regulating the internal environment and working with other body systems.
The muscular system consists of three types of muscles - skeletal, smooth, and cardiac. Skeletal muscles are voluntary and attach to bones to enable movement. Smooth muscles line organs and blood vessels to regulate movement within the body. Cardiac muscle is only found in the heart to pump blood throughout the body. Muscles contract and relax to perform functions like movement, stability, protection, and temperature regulation. A healthy diet and exercise are important to maintain strong, healthy muscles. The muscular system also helps maintain homeostasis by regulating temperature and transporting oxygen and waste throughout the body.
Charles Darwin developed the theory of evolution by natural selection after observing variations in species on the Galapagos Islands. He noticed that tortoises, finches, and other animals had adapted to their environments over time through traits that improved their chances of survival, such as tortoises developing different neck lengths corresponding to the plants available on each island. Darwin's theory explained how evolution can occur gradually through natural selection acting upon heritable variations that increase organisms' likelihood of surviving and reproducing.
DNA contains genes that provide instructions for making proteins. DNA has a double helix structure with two strands coiled around each other. Each strand contains repeating sequences of nucleotides with one of four nitrogen bases (A, T, C, G). RNA is similar but single-stranded and helps carry instructions from DNA in the nucleus to the cell's protein-making machinery. Mutations can occur during DNA replication, resulting in changes to genes that may cause genetic disorders or beneficial trait variations.
The document discusses the concepts of inheritance, genes, alleles, genotypes, phenotypes, and patterns of inheritance. It explains that chromosomes contain genes which control traits, and that offspring inherit genes from both parents. Genotypes are an organism's combination of alleles, while phenotypes are the observable traits. Dominant and recessive alleles can interact in different inheritance patterns like incomplete dominance, codominance, multiple alleles, and polygenic inheritance. An organism's environment and multiple genes can also influence phenotypes. Some disorders are caused by recessive or sex-linked genes.
Gregor Mendel performed experiments with pea plants from 1856 to 1863 to study heredity. He found that pea plants have traits such as flower color and seed shape that are inherited. Through controlled breeding experiments involving over 28,000 pea plants, Mendel discovered that traits are passed to offspring through discrete factors, now known as genes, and that some traits are dominant over recessive traits. His findings disproved the prevailing theory of blending inheritance and established the basic principles of genetics.
Asexual reproduction allows organisms to reproduce without meiosis and fertilization, resulting in offspring that are genetically identical to the parent. It occurs through various methods like fission, budding, regeneration, vegetative reproduction, and cloning. Fission involves a prokaryotic cell splitting into two identical daughter cells. Budding occurs when an outgrowth from the parent develops into a new individual. Regeneration involves regrowing a new individual from a fragment of the parent. Vegetative reproduction is seen in plants that produce new individuals from stems, leaves, or other vegetative plant structures. Cloning produces genetically identical copies in a laboratory setting. Asexual reproduction allows for rapid population growth without locating a mate. [/SUMMARY]
Sexual reproduction involves the combination of genetic material from two parent cells to form a new cell. It occurs through meiosis which produces haploid sex cells with half the number of chromosomes and through fertilization where an egg and sperm join. This maintains the diploid number of chromosomes and generates genetic variation in offspring, providing advantages for adaptation and selective breeding.
This document discusses the levels of organization of living things from atoms to organisms. It begins by explaining that all matter is made of atoms which combine to form molecules and cells. Cells make up unicellular and multicellular organisms. Unicellular organisms consist of a single cell and can be prokaryotic or eukaryotic. Multicellular organisms are made of many eukaryotic cells that differentiate and organize into tissues and organs to carry out specific functions needed for organism survival.
The cell cycle consists of interphase and the mitotic phase. Interphase includes three stages (G1, S, G2) where the cell grows and duplicates its DNA. The mitotic phase includes mitosis, where the cell nucleus and chromosomes divide, and cytokinesis, where the cell cytoplasm divides to form two daughter cells each with the same genetic material as the original cell. Cell division through the cell cycle enables growth, development, replacement of old/damaged cells, and repair of injuries in multicellular organisms.
Cellular respiration is a series of chemical reactions that convert energy from food into a usable form called ATP. It takes place in two steps - glycolysis in the cytoplasm breaks down glucose, producing some ATP and precursor molecules, while the second step in mitochondria uses oxygen to break down these precursors and produce much more ATP. Fermentation is an alternative pathway used without oxygen to produce less ATP. Photosynthesis converts light energy, carbon dioxide, and water into glucose and oxygen through reactions in chloroplasts.
The document discusses different types of transport across cell membranes. It explains that cell membranes are semipermeable and allow certain materials to pass through. Small molecules can pass through passively via diffusion, osmosis, or facilitated diffusion without using energy. Larger molecules and substances moving against a concentration gradient require active transport which uses the cell's energy. Other processes like endocytosis and exocytosis allow larger particles and molecules to enter or exit cells.
Cells come in a wide range of sizes, with human egg cells being the largest at around 1/10 the size of a period at the end of a sentence in 12 point font. Bacteria are much smaller, with around 8,000 bacteria able to fit inside a single human egg cell. All cells share some common traits, including a cell membrane that regulates interactions between the cell and its environment, cytoplasm located inside the cell, and hereditary material called DNA that controls the cell. However, plant, bacteria, and other cell types can also contain additional structures like a cell wall, chloroplasts, or flagella.
3. Quadratic
Function
non linear function
that can be written
in standard form,
y = ax2 + bx + c
4. Quadratic Parabola
Function U-shaped graph that
non linear function a quadratic function
that can be written makes
in standard form,
y = ax2 + bx + c
5. Quadratic Parabola
Function U-shaped graph that
non linear function a quadratic function
that can be written makes
in standard form,
y = ax2 + bx + c
6. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes
in standard form,
y = ax2 + bx + c
7. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes
in standard form,
y = ax2 + bx + c
8. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
9. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
10. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
11. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
Parent Quadratic Function
the most basic quadratic equation, y = x2
12. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
Axis of Symmetry
the line that passes through the vertex and
divides the parabola in two symmetrical
parts. The a of s of y = x2 is x=0
Parent Quadratic Function
the most basic quadratic equation, y = x2
13. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
Axis of Symmetry
the line that passes through the vertex and
divides the parabola in two symmetrical
parts. The a of s of y = x2 is x=0
Parent Quadratic Function
the most basic quadratic equation, y = x2
14. Quadratic Parabola Vertex
Function U-shaped graph that the lowest or highest
non linear function a quadratic function point on a parabola
that can be written makes The vertex of the
in standard form, parent equation
y = ax2 + bx + c y = x2 is (0,0)
Axis of Symmetry
the line that passes through the vertex and
divides the parabola in two symmetrical
parts. The a of s of y = x2 is x=0
Parent Quadratic Function
the most basic quadratic equation, y = x2
17. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
18. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
19. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
★Step 4:
Compare the
graphs (vertex,
axis of symmetry,
vertical stretch)
22. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
23. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
24. ★Step 1: Example 1
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
★Step 4:
Compare the
graphs (vertex,
axis of symmetry,
vertical stretch)
25. Graph y = 1/2x2. Compare the
Example 2 graph with the graph of y = x2
26. Graph y = 1/2x2. Compare the
★Step 1: Example 2 graph with the graph of y = x2
Make a table of
values
27. Graph y = 1/2x2. Compare the
★Step 1: Example 2 graph with the graph of y = x2
Make a table of
values
★Step 2:
Plot the points
from the tables
28. Graph y = 1/2x2. Compare the
★Step 1: Example 2 graph with the graph of y = x2
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
29. Graph y = 1/2x2. Compare the
★Step 1: Example 2 graph with the graph of y = x2
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
★Step 4:
Compare the
graphs (vertex,
axis of symmetry,
vertical stretch)
32. ★Step 1: Example 2
Make a table of
values
★Step 2:
Plot the points
from the tables
33. ★Step 1: Example 2
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
34. ★Step 1: Example 2
Make a table of
values
★Step 2:
Plot the points
from the tables
★Step 3:
Draw a smooth
curve through
the points
★Step 4:
Compare the
graphs (vertex,
axis of symmetry,
vertical stretch)
35. Comparing to
y=x 2
When |a|>1, then there is a vertical stretch,
by a factor of a.
When |a|<1, then there is a vertical shrink,
by a factor of a.
When a is negative, whether a>1 or a<1,
then there is a reflection in the x-axis.
40. Comparing to
y=x 2
When |a|>1, then there is a vertical stretch, by a factor of a.
When |a|<1, then there is a vertical shrink, by a factor of a.
When a is negative, whether a>1 or a<1, then there is a reflection in the x-axis.
When c is positive, then there is a vertical
translation up c units.
When c is negative, then there is a vertical
translation down c units.