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3. Functions

Transformations

3. Functions

Transformations

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Intro to Transformations

Intro to Transformations

Reflections of Functions

Reflections of Functions

Reflections of Functions Example 1

Reflections of Functions Example 1

Shifts of Functions

Shifts of Functions

Graphs of Shifted & Reflected Functions

Graphs of Shifted & Reflected Functions

Graphs of Shifted & Reflected Functions Example 1

Graphs of Shifted & Reflected Functions Example 1

Stretches & Shrinks of Functions

Stretches & Shrinks of Functions

Stretches & Shrinks of Functions Example 1

Stretches & Shrinks of Functions Example 1

Domain & Range of Transformed Functions

Domain & Range of Transformed Functions

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Format for Vertical Shifts of Graphs of Functions

Format for Vertical Shifts of Graphs of Functions

Vertical Shift of a Graph

Vertical Shift of a Graph

Horizontal Shift of a Graph

Horizontal Shift of a Graph

Format for Horizontal Shifts of Graphs of Functions

Format for Horizontal Shifts of Graphs of Functions

M8: Reflection Across the y-Axis

M8: Reflection Across the y-Axis

Format for Reflection of Function Graphs Across the y-Axis

Format for Reflection of Function Graphs Across the y-Axis

Reflection Across the x-Axis

Reflection Across the x-Axis

Format for Reflection of Function Graphs Across the x-Axis

Format for Reflection of Function Graphs Across the x-Axis

Vertical Stretch or Compression of the Graph of a Function

Vertical Stretch or Compression of the Graph of a Function

Format for the Vertical Stretch or Compression of the Graphs of Functions

Format for the Vertical Stretch or Compression of the Graphs of Functions

Format for the Vertical Stretch or Compression of the Graphs of Functions

Format for the Vertical Stretch or Compression of the Graphs of Functions

Horizontal Stretch or Compression of the Graph of a Function

Horizontal Stretch or Compression of the Graph of a Function

Combined Shifting of a Function

Combined Shifting of a Function

Write an Equation From a Description of Transformations

Write an Equation From a Description of Transformations

Applying Combinations of Transformations

Applying Combinations of Transformations

Perform Transformations on a Given Graph

Perform Transformations on a Given Graph

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Practice this topic

Multiple Choice

Written below (green dotted curve) is a graph of the function $f\left(x\right)=\sqrt{x-2}$ . If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x)?

Multiple Choice

The green dotted line in the graph below represents the function $f\left(x\right)$ . The blue solid line represents the function $g\left(x\right)$ , which is the function $f\left(x\right)$ after it has gone through a shift transformation. Find the equation for $g\left(x\right)$ .

Multiple Choice

The green dotted curve below is a graph of the function $f\left(x\right)$ . Find the domain and range of $g\left(x\right)$ (the blue solid curve), which is a transformation of $f\left(x\right)$ .

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Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

g(x) = -f(x) +3

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

g(x) = -f(2x) - 1

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=3|x|

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g.

g(x) = f(x) - 1

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g.

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=2/3|x|

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g.

g(x) = f(x-1)+2

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g. g(x) = f(x + 1) − 2

Textbook Question

Graph each function. See Examples 1 and 2. g(x)=(1/2)x^2

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g. g(x) = f(-x)

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-(1/2)x^2

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g.

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g. g(x) = -f(x)+1

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-3|x|

Textbook Question

In Exercises 17-32, use the graph of y = f(x) to graph each function g.

Textbook Question

Graph each function. See Examples 1 and 2. h(x)=|-(1/2)x|

Textbook Question

Graph each function. See Examples 1 and 2. h(x)=√(4x)

Textbook Question

In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = f(x)+2

Textbook Question

Graph each function. See Examples 1 and 2. ƒ(x)=-√-x

Textbook Question

In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = f(x+2)

Textbook Question

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (5, -3)

Textbook Question

In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = -(1/2)f(x+2)

Textbook Question

Plot each point, and then plot the points that are symmetric to the given point with respect to the (a) x-axis, (b) y-axis, and (c) origin. (-4, -2)

Textbook Question

In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = -½ ƒ ( x + 2) —2

Textbook Question

In Exercises 33-44, use the graph of y = f(x) to graph each function g. g(x) = (1/2)f(2x)

Textbook Question

Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4. y=x^2+5

Textbook Question

In Exercises 45-52, use the graph of y = f(x) to graph each function g. g(x) = -f(x-1) + 1

Textbook Question

Without graphing, determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. See Examples 3 and 4. x^2+y^2=12

Textbook Question

In Exercises 45-52, use the graph of y = f(x) to graph each function g. g(x) = -f(x + 1) − 1

Textbook Question

In Exercises 45-52, use the graph of y = f(x) to graph each function g. g(x)=2f(x-1)

Textbook Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = x² - 2

Textbook Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (x − 2)²

Textbook Question

In Exercises 55–59, use the graph of to graph each function g.

g(x) = f(x + 2) + 3

Textbook Question

In Exercises 55–59, use the graph of to graph each function g.

Textbook Question

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. g(x) = x^2 + 2

Textbook Question

In Exercises 60–63, begin by graphing the standard quadratic function, f(x) = x^2. Then use transformations of this graph to graph the given function. r(x) = -(x + 1)^2

Textbook Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. g(x) = (1/2)(x − 1)²

Textbook Question

In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x + 3)

Textbook Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = (1/2) (x − 1)² – 1

Textbook Question

In Exercises 64–66, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. r(x) = 2√(x + 2)

Textbook Question

In Exercises 53-66, begin by graphing the standard quadratic function, f(x) = x². Then use transformations of this graph to graph the given function. h(x) = -2(x+2)²+1

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=x^2+2

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √(x+1)

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=-√(x + 1)

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order. y = | x - 1 | y = x^2 - 4 How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. h(x)=-(x+1)^3

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x) = √(x+1)-1

Textbook Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)-1

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=-3(x-2)^2+1

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x|+3

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x+3|

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=2√x+1

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = |x + 3| - 2

Textbook Question

Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=3√x-2

Textbook Question

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = 2|x+3|

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2

Textbook Question

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Textbook Question

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = x³-3

Textbook Question

Describe how the graph of each function can be obtained from the graph of ƒ(x) = |x|. g(x) = -|x|

Textbook Question

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. g(x) = (x − 3)^3

Textbook Question

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x). across the x-axis

Textbook Question

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. h(x) = -x³

Textbook Question

Let ƒ(x) = 3x -4. Find an equation for each reflection of the graph of ƒ(x). across the y-axis

Textbook Question

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(a) y = ƒ(x) +3

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(b) y = ƒ(x-2)

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(c) y = ƒ(x+3) - 2

Textbook Question

In Exercises 95-106, begin by graphing the standard cubic function, f(x) = x³. Then use transformations of this graph to graph the given function. r(x) = (x − 2)³ +1

Textbook Question

The graph of a function ƒ is shown in the figure. Sketch the graph of each function defined as follows.

(d) y = |ƒ(x)|

Textbook Question

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛x+2

Textbook Question

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = ∛(x-2)

Textbook Question

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x-2)

Textbook Question

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. g(x) = (1/2)∛(x+2) - 2

Textbook Question

In Exercises 107-118, begin by graphing the cube root function, f(x) = ∛x. Then use transformations of this graph to graph the given function. ∛(-x-2)

Textbook Question

In Exercises 81–94, begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -|x + 4| +2

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