0. Functions

Graphing

In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.

y = (−2x)²/³

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

d. 𝔂 = ƒ(2x + 1)

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

e. 𝔂 = ƒ( x ) - 4

3

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=x^2$ into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=\text{ƒ}(2x-4)$

How do you obtain the graph of $y=-3f\(\left\)(x\(\right\))$ from the graph of $y=f\(\left\)(x\(\right\))$?

Shifting a graph Use a shift to explain how the graph of $\text{ƒ}\left(x\right)=\sqrt{-x^2+8x+9}$ is obtained from the graph of $g\left(x\right)=\sqrt{25-x^2}$ . Sketch a graph of ƒ.

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = −√x Right 3

How do you obtain the graph of $y=f\(\left\)(3x\(\right\))$ from the graph of $y=f\(\left\)(x\(\right\))$?

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = (1/2)(x + 1) + 5 Down 5, right 1

Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).

a. 𝔂 = ƒ(x - 5)

Shifting and Scaling Graphs

Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.

f. Compress horizontally by a factor of 5

Graph the functions in Exercises 37–56.

y = (x + 2)³/² + 1

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=x^2$ into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=6\text{ƒ}\left(\frac{x-2}{3}\right)+1$

Graph the functions in Exercises 37–56.

y = (1/x²) − 1

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=\sqrt{x}$  into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=2\text{ƒ}(2x-1)$