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1.2: Basic Classes of Functions

https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/01%3A_Functions_and_Graphs_(Precalculus_Review)/1.02%3A_Basic_Classes_of_Functions

Figure

$\PageIndex{9}$ : (a) For

$c>0$ , the graph of

$y=f(x)+c$ is a vertical shift up

$c$ units of the graph of

$y=f(x)$ . (b) For

$c>0$ , the graph of

$y=f(x)−c$ is a vertical shift down ...Figure

$\PageIndex{9}$ : (a) For

$c>0$ , the graph of

$y=f(x)+c$ is a vertical shift up

$c$ units of the graph of

$y=f(x)$ . (b) For

$c>0$ , the graph of

$y=f(x)−c$ is a vertical shift down c units of the graph of

$y=f(x)$ . For

$c>0$ , the graph of

$f(x+c)$ is a shift of the graph of

$f(x)$ to the left

$c$ units; the graph of

$f(x−c)$ is a shift of the graph of

$f(x)$ to the right

$c$ units.

1.3: Basic Classes of Functions

https://math.libretexts.org/Courses/SUNY_Geneseo/Math_221_Calculus_1/01%3A_Functions_and_Graphs/1.03%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.2: Basic Classes of Functions

https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_1_(Sklar)/01%3A_Functions_and_Graphs/1.02%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.3: Basic Classes of Functions

https://math.libretexts.org/Courses/Laney_College/Math_3A%3A_Calculus_1_(Fall_2022)/01%3A_Functions_and_Graphs/1.03%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.2: Basic Classes of Functions

https://math.libretexts.org/Courses/Reedley_College/Calculus_I_(Casteel)/01%3A_Functions_and_Graphs/1.02%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.2: Basic Classes of Functions

https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_hdagnew@ucdavis.edu/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2F1%3A_Functions_and_Graphs_(Review)/Courses%2F%2FRemixer_University%2F%2FUsername%3A_hdagnew@ucdavis.edu%2F%2FMonroe2%2F%2F1%3A_Functions_and_Graphs_(Review)%2F%2F1.2%3A_Basic_Classes_of_Functions

For

$c>0$ , the graph of

$f(x+c)$ is a shift of the graph of

$f(x)$ to the left

$c$ units; the graph of

$f(x−c)$ is a shift of the graph of

$f(x)$ to the right

$c$ units. For example, the...For

$c>0$ , the graph of

$f(x+c)$ is a shift of the graph of

$f(x)$ to the left

$c$ units; the graph of

$f(x−c)$ is a shift of the graph of

$f(x)$ to the right

$c$ units. For example, the graph of the function

$f(x)=3x^2$ is the graph of

$y=x^2$ stretched vertically by a factor of 3, whereas the graph of

$f(x)=x^2/3$ is the graph of

$y=x^2$ compressed vertically by a factor of

$3$ (Figure

$\PageIndex{8}$ ).

1.7: Transformations

https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/01%3A_Relations_and_Functions/1.07%3A_Transformations

This section covers transformations of functions, including translations, reflections, stretches, and compressions. It explains how to apply these transformations to function graphs and how changes in...This section covers transformations of functions, including translations, reflections, stretches, and compressions. It explains how to apply these transformations to function graphs and how changes in function equations affect their graphs. The section provides examples and visual aids to illustrate vertical and horizontal shifts, reflections over the axes, and scaling of graphs. These concepts help understand how functions can be manipulated to model different real-world scenarios.

1.2: Basic Classes of Functions

https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/1%3A_Functions_and_Graphs_(Review)/1.2%3A_Basic_Classes_of_Functions

For

$c>0$ , the graph of

$f(x+c)$ is a shift of the graph of

$f(x)$ to the left

$c$ units; the graph of

$f(x−c)$ is a shift of the graph of

$f(x)$ to the right

$c$ units. For example, the...For

$c>0$ , the graph of

$f(x+c)$ is a shift of the graph of

$f(x)$ to the left

$c$ units; the graph of

$f(x−c)$ is a shift of the graph of

$f(x)$ to the right

$c$ units. For example, the graph of the function

$f(x)=3x^2$ is the graph of

$y=x^2$ stretched vertically by a factor of 3, whereas the graph of

$f(x)=x^2/3$ is the graph of

$y=x^2$ compressed vertically by a factor of

$3$ (Figure

$\PageIndex{8}$ ).

1.3: Basic Classes of Functions

https://math.libretexts.org/Courses/City_University_of_New_York/Calculus_I_(CUNY)/01%3A_Functions_and_Graphs/1.03%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.2: Basic Classes of Functions

https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/01%3A_Functions_and_Graphs/1.02%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

1.3: Basic Classes of Functions

https://math.libretexts.org/Courses/Chabot_College/MTH_1%3A_Calculus_I/01%3A_Functions_and_Graphs/1.03%3A_Basic_Classes_of_Functions

We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define gene...We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.

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