Exam 6: Transformations of Functions and Their Graphs

The graph of a function $f$ has been shifted down 4 units, shifted 7 units to the right, and then stretched vertically by a factor of 9 . The new graph is produced by a function $g$ . Find a formula for $g$ in terms of $f$ .

Some of the values of $f(x)$ are given in the following table. If $f(x)$ is an odd function, what is $f(-3)$ ? x 1 2 3 4 5 f(x) 1 0 -1 0 3

The following table gives values for $c(x)$ , the cost per foot of drilling a well at a depth of $x$ feet. The prices given are for wells drilled on a certain ranch in Texas. The ranch sits above a dense layer of bedrock. At what depth do you estimate that the bedrock layer begins?

Which of the following functions are even?

Let $f(x)=-3 x^{2}+3$ . Write a formula for the following transformations of $f$ . A) $g(x)=2 f(x)$ B) $h(x)=3 f(x-1)$ C) $m(x)=-f(x)+4$

The point $(-4,5)$ lies on the graph of $f$ . What point must lie on the graph of $3 f\left(\frac{x}{5}\right)$ ?

Let $V$ represent the volume of a box with square base of side length $x$ and height 8 . By what percent does the volume increase if the side length is increased by $15 \%$ ?

Let $V(x)$ represent the volume in cubic feet of a box with square base of side length $x$ feet and height 7 feet. Let $T(x)$ represent the volume in cubic inches of a box with square base of side length $x$ inches and height 84 inches. Which of the following is the correct formula for $T(x)$ in terms of $V(x)$ ?

The function $f(x)$ has odd symmetry and the function $g(x)$ has even symmetry. Is the function $h(x)=2 g(x+1)$ odd, even, or neither?

If the graph of $f(x)$ is: Then the graph: Represents:

The function $f(x)$ has odd symmetry and the function $g(x)$ has even symmetry. Is the function $h(x)=2 f(x)$ odd, even, or neither?

The function $g(x)$ is obtained from $f(x)$ by a single transformation. Use the tables below to find a formula for $g(x)$ in terms of $f(x)$ .

The domain of $f(x)$ is $-2 \leq x \leq 6$ and the range is $-4 \leq f(x) \leq 5$ . What is the domain and range of $-2 f(x)$ ?

Educational psychologists suggest that a child's intellectual capacity, $I$ , can be represented as a function of time, $t$ , in months, as follows: $I=f(t)=a\left(1-e^{-b t}\right)$ Where $a>0, b>0$ are constants. What is the effect on the graph of $I$ of increasing the value of $a$ ?

Let $f(x)=e^{x}$ . Which of the following describe how the function $g(x)=4 e^{6(x-3)}$ transforms the graph of $f$ ?

Below is a graph of the function $f(x)$ . If you were to sketch a graph of $y=f(-x)+2$ , what point would the point $(-2,1)$ correspond to?

Suppose $f(x)$ is an odd function. Determine whether $g(x)=f(2 x)+16$ is odd, even, or neither. Write "cannot tell" if there is not enough information to determine.

The graph of $g(x)$ contains the point $(1,5)$ . What is the corresponding point on the graph of $y=-4 g(x)+4$ ?

The graph of $f(x)$ contains the point $(4,-3)$ . What corresponding point must be on the graph of $g(x)=-3 f(x-10)$ ?

The domain of $f(x)$ is $-16 \leq x \leq 16$ and the range is $0 \leq f(x) \leq 11$ . What is the domain and range of $g(x)=f(-4 x)$ ?