# MTH 111–112 Supplement

## Section1.1Graph Transformations

### Example1.1.1.

The table below defines the functions $$f\text{,}$$ $$g\text{,}$$ and $$h\text{.}$$ Express $$g(x)$$ and $$h(x)$$ in terms of $$f\text{.}$$
 $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$f(x)$$ $$8$$ $$6$$ $$4$$ $$2$$ $$0$$ $$-1$$ $$-2$$ $$g(x)$$ $$-8$$ $$-6$$ $$-4$$ $$-2$$ $$0$$ $$1$$ $$2$$ $$h(x)$$ $$5$$ $$3$$ $$1$$ $$1$$ $$-3$$ $$-4$$ $$-5$$
$$g(x)=-f(x)$$ and $$h(x)=f(x)-3\text{.}$$

### Example1.1.2.

#### (a)

If $$f(x)=x^2$$ and $$g(x)=2x^2+5\text{,}$$ express $$g(x)$$ in terms of $$f\text{.}$$
$$g(x)=2f(x)+5$$

#### (b)

If $$f(x)=x^2$$ and $$h(x)=(x+5)^2-3\text{,}$$ express $$h(x)$$ in terms of $$f\text{.}$$
$$h(x)=f(x+5)-3$$

### ExercisesExercises

#### One Function in Terms of Another.

In Exercises 1–4, the table below defines the functions $$f\text{,}$$ $$g\text{,}$$ $$h\text{,}$$ $$k\text{,}$$ and $$l\text{.}$$
 $$x$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$f(x)$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$g(x)$$ $$4$$ $$3$$ $$2$$ $$1$$ $$0$$ $$h(x)$$ $$0$$ $$-1$$ $$-2$$ $$-3$$ $$-4$$ $$k(x)$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$l(x)$$ $$0$$ $$3$$ $$6$$ $$9$$ $$12$$
##### 1.
Express $$g(x)$$ in terms of $$f$$ and describe how the graph of $$y=f(x)$$ can be transformed into the graph of $$y=g(x)\text{.}$$
##### 2.
Express $$h(x)$$ in terms of $$f$$ and describe how the graph of $$y=f(x)$$ can be transformed into the graph of $$y=h(x)\text{.}$$
##### 3.
Express $$k(x)$$ in terms of $$f$$ and describe how the graph of $$y=f(x)$$ can be transformed into the graph of $$y=k(x)\text{.}$$
##### 4.
Express $$l(x)$$ in terms of $$f$$ and describe how the graph of $$y=f(x)$$ can be transformed into the graph of $$y=l(x)\text{.}$$

#### 5.

The second row in the table below gives values for the function $$f\text{.}$$ Complete the rest of the table. If you don’t have sufficient information to fill in some of the cells, leave those cells blank.
 $$x$$ $$-4$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$f(x)$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$\frac{1}{2}x$$ $$-2f(x)$$ $$f(x)+5$$ $$f(x+2)$$ $$f\mathopen{}\left( \frac{1}{2}x \right)\mathclose{}$$ $$f(2x)$$ $$f(x-3)$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$ $$\phantom{-0}$$

#### Find the Transformations.

In Exercises 6–9, first write $$g(x)$$ in terms of $$f\text{.}$$ Then compose a sequence of transformations that will transform the graph of $$y=f(x)$$ into the graph of $$y=g(x)\text{.}$$
##### 6.
$$f(x)=\sqrt{x}\qquad g(x)=\frac{\sqrt{x-7}}{4}$$
##### 7.
$$f(x)=\frac{1}{x}\qquad g(x)=\frac{2}{x}+3$$
##### 8.
$$f(x)=x^2\qquad g(x)=-4\mathopen{}\left( \frac{1}{2}x-5 \right)^2\mathclose{}+3$$
##### 9.
$$f(x)=\sqrt[3]{x}\qquad g(x)=\frac{1}{2}\sqrt[3]{10x+30}-6$$

#### Sketch Transformations.

In Exercises 10–13, use the provided graph of $$y=f(x)$$ to sketch a graph of each given function.
##### 10.
$$k_1(x)=f(2x)$$
##### 11.
$$k_2(x)=2f(-2x)-1$$
##### 12.
$$k_3(x)=-2f(2x+4)$$
##### 13.
$$k_4(x)=f\mathopen{}\left(\frac{1}{2}x\right)\mathclose{} +2$$