College Algebra 5.1
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This document discusses exponential functions and their graphs. It covers the basic exponential function f(x) = a^x where a is a positive number. Examples are given of evaluating basic exponential functions at different values of x. The one-to-one property of exponential functions is demonstrated. The graph of the basic exponential function f(x) = 2^x is shown and its asymptote is identified as the line y = 0. Shifting and reflecting the graph are discussed, and it is noted how the asymptote changes with shifting.
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3. The vertical line test is introduced as a way to visually check if a relation is a function by seeing if any vertical line passes through more than one point.
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- Identity functions where the output is equal to the input.
- Polynomial functions defined by expressions involving powers of the variable.
- Absolute value functions which output the absolute value of the input.
- Square root functions which output the square root of the input.
- Rational functions where the output is a rational expression involving the input.
It also discusses the domain and range of these different function types.
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The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
Grph quad fncts
The document discusses graphing quadratic functions. It defines a quadratic function as f(x) = ax^2 + bx + c where a, b, and c are real numbers and a is not equal to 0. The graph of a quadratic function is a parabola that is symmetrical about an axis. When the leading coefficient a is positive, the parabola opens upward and the vertex is a minimum. When a is negative, the parabola opens downward and the vertex is a maximum. Standard forms for quadratic functions and methods for finding characteristics like the vertex, axis of symmetry, and x-intercepts from the equation are also presented.
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An asymptote is an imaginary line that a graph approaches but never touches as it goes to infinity. The document provides examples of graphs with rational functions that have vertical and horizontal asymptotes. It also discusses the square root function and how it is the inverse of the quadratic function. The square root function is only defined for non-negative inputs. Changing the parameters a and b in the square root function changes the shape and orientation of the graph.
Solution 3
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
Algebra 2. 9.16 Quadratics 2
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
119 Powerpoint 2.2
This document discusses functions and how to work with them. It defines functions and explains that a function relates each input to a single unique output. It provides examples of relations that are and are not functions. It also discusses how to evaluate functions, find domains, use function notation like f(x), and calculate difference quotients.
Unit 2.6
The document discusses rational functions and their graphs. It defines rational functions as the ratio of two polynomial functions. Key aspects covered include:
- The domain of a rational function
- Transformations of the reciprocal function 1/x and how they affect its graph
- Vertical and horizontal asymptotes of rational functions
- End behavior, intercepts, and characteristics of graphs of general rational functions
- Examples of finding asymptotes and graphing specific rational functions
Module 2 quadratic functions
1. This module covers quadratic functions and how to graph them. Students will learn to identify the vertex, axis of symmetry, and direction of opening of quadratic graphs.
2. Key aspects covered include how changing coefficients a, h, and k affect the width, translation, and vertical shift of the graph. When a increases, the graph narrows; changing h translates the graph left or right; and changing k shifts the graph up or down.
3. The steps for graphing a quadratic function are finding the vertex, axis of symmetry, direction of opening, and making a table of values to plot points.
Mathematics 9 Quadratic Functions (Module 2)
This learner's module will discuss or talk about the Graph of Quadratic Functions. It will also discuss on how to draw the Graph of Quadratic Functions using the vertex, axis of symmetry, etc.
Fun37
1) Rational functions are quotients of two polynomial functions. The parent rational function is f(x) = 1/x.
2) Graphs of rational functions can have vertical asymptotes where the function is undefined, horizontal asymptotes as constant lines the function approaches, and sometimes slant asymptotes.
3) Vertical asymptotes occur where the denominator is zero. Horizontal asymptotes occur depending on the relative degrees of the numerator and denominator: if the numerator degree is less than the denominator degree the horizontal asymptote is y=0, if the degrees are equal the horizontal asymptote is the quotient, and if the numerator degree is greater there is no horizontal asymptote.
Functions for Grade 10
This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.
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College Algebra 5.1
- 1. 5.1 Exponential Functions and Their Graphs A. Basic Exponential Function (increasing and decreasing) B. One-to-one property C. Graph of Basic Exponential Function including Asymptote D. Shifting Graph of Basic Exponential Function (including asymptote)
- 3. Suppose f(x) = 2 x . Find f(4).
- 5. Suppose f(x) = (1/3) x . Find f(4).
- 9. You try:
- 11. Let’s compare that shape to f(x)=4 x x f(x)=4 x (x,y) 0 1 4 0 =1 4 1 = (0,1) ( -1 -2 4 -1 = 4 -2 = -3 2 4 -3 = 4 2 =
- 16. You try: Shift f(x) = 2 x to the left 2 units. Gve the equation, the graph, and the asymptote.
- 17. You try: Reflect f(x) = 2 x over the x-axis. Give the equation, the graph, and the asymptote.