Graphing Quadratic Functions
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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.
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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.
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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.
April 3, 2014
The document provides examples and explanations for graphing quadratic functions. It begins with an overview of how the a, b, and c values in the quadratic function y=ax2 + bx + c impact the graph. Examples are then worked through step-by-step to show how to find the axis of symmetry, vertex, y-intercept, and additional points to graph the function. An application example models the height of a basketball shot as a quadratic function to find the maximum height and time to reach it. The document concludes with a check your understanding example modeling the height of a dive.
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The document discusses how to graph quadratic equations in the form of y = ax^2 + bx + c. It states that the graphs are parabolas with a vertex and center line. To graph a parabola, one finds the vertex, another point such as the y-intercept, reflects that point across the center line, and finds the x-intercept to complete the parabola.
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The document discusses three forms of quadratic equations - standard, vertex, and intercept form - and how to graph each. It provides the key characteristics and steps to graph quadratics in each form. It also discusses two special factoring patterns for standard form and how to convert between the three forms.
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This document discusses the behavior and key characteristics of quadratic functions and their graphs (parabolas). It begins by defining quadratic functions, expressions, and equations. It then explains that a quadratic function has two variables, x and y, where x is the independent variable and y is the dependent variable.
The document goes on to discuss several important aspects of parabolas including the vertex (point of minimum or maximum), axis of symmetry, zeros (where the graph crosses the x-axis), and point of origin. It then analyzes the graphs of various quadratic functions of the form y=ax^2, observing that when a is positive the graph opens upward and the opening gets narrower as a increases, and when a
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G9_Q2_W2_L2_GraphsofQUadraticFunction.pdf
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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.
Solving Quadratics
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Graphing Quadratics
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.
Graphing Quadratics
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.
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This document discusses various concepts related to vectors and 3D geometry including dot products, cross products, planes, lines, and their relationships. Dot products can be used to find the angle between vectors and determine if vectors are perpendicular. Cross products give a vector perpendicular to both input vectors. Plane equations can be defined using a point and normal vector, three points, or two vectors in the plane. Lines are defined by two points or a point and direction vector. The intersection of planes and lines, parallelism, and distances between lines and points and planes are also covered.
Quadratic function in standard form (y = ax^2 +bx + c
Graphing quadratic functions, identify maximum and minimum values, how to identify the lines of symmetry, how to find the vertex,
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- 1. Basics of Graphing Quadratic Functionsadapted from hisema01
- 2. What is a Quadratic Function? A function with the form y = ax2 + bx + c where a 0. The graph is U-shaped and is called a parabola.
- 3. Why Study Quadratics? Many things we see every day are modeled by quadratic functions. What are some examples? Water in a drinking fountain The McDonald’s Golden Arches The path of a basketball
- 4. Quadratic Vocab The lowest or highest point is called the vertex. The axis of symmetry is a vertical line through the vertex.
- 5. Effects of “a” Standard Form: y = ax2 + bx + c Just like with absolute value functions: If a > 0 (+), the parabola opens up If a < 0 (-), the parabola opens down
- 6. Effects of “a” Standard Form: y = ax2 + bx + c Just like with absolute value functions: If |a| < 1, the parabola is wider than y = x2 because it’s vertically shrunk If |a| > 1, the parabola is narrower than y = x2 because it’s vertically stretched.
- 7. Effects of “c” Standard Form: y = ax2 + bx + c Just like with absolute value functions: If c < 1, the parabola is shifted down c units If c > 1, the parabola is shifted up c units
- 8. Finding the Vertex The x-coordinate of the vertex is − 𝑏 2𝑎 To find the y, plug the x-coordinate into the equation. Axis of symmetry is the line x = − 𝑏 2𝑎
- 9. Graphing in Standard Form 1. Find and plot the vertex: − −8 2 2 = 2 𝑦 = 2(2)2−8 2 + 6 = −2 2. Draw the axis of symmetry: x = 2 Example: y = 2x2 – 8x + 6
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