Exam 6: Transformations of Functions and Their Graphs

The graph of a function $f$ has been shifted down 2 units, shifted 5 units to the right, and then stretched vertically by a factor of 8 , and compressed horizontally by a factor of 3. The new graph is produced by a function $g$ . Find a formula for $g$ in terms of $f$ .

The following table gives values for the functions $f$ and $g$ . Which of the following statements are true?

The point $(-4,4)$ lies on the graph of $f$ . If the graph of $f$ is compressed vertically by a factor of $\frac{1}{7}$ and compressed horizontally by a factor of $\frac{1}{11}$ , what point must lie on the transformed graph?

Let $P(n)$ be the profit made by a water park in terms of the number of people, $n$ , visiting the park each day. If $n_{0}$ people attended the park yesterday, what is meant by the expression $P\left(n_{0}\right)+10$ ?

Let 5 and -4 be the zeros of the quadratic function $f(x)$ . What are the zeros of the function $g(x)=f(5 x)$ ?

Given the graph of $y=f(x)$ below, find a possible formula for $f(x)$ .

Describe the effect of the transformation $f\left(\frac{x}{7}\right)+6$ on the graph of $f(x)$ .

The following is the graph of $f(x)$ . Sketch the graph of $f(-x)$ .

Let $g(x)=\frac{1}{x^{2}}$ . The domain of the formula that transforms the graph of $g$ by shifting it 8 units to the right, reflecting it about the $x$ -axis, and then shifting it up 4 units is all real numbers except $x=$ ------------

Let $g(x)=\frac{1}{x^{2}}$ . Which of the following formulas transforms the graph of $g$ by shifting it 5 units to the right, reflecting it about the $x$ -axis, and then shifting it up 3 units?

Graph $g(x)=-2 f(x)$ if $f(x)=e^{x}$ .

The graphs of $f$ and $g$ are shown at the right. Which of the following gives a formula for $g(x)$ in terms of $f(x)$ ?

How are the graphs of $y=g(x)=e^{7 x}$ and $y=g(-x)$ related?

The graph of $g(x)$ is the graph of $f(x)$ after it has been vertically stretched or shrunk. The point $(3,12)$ lies on the graph of $f(x)$ . The corresponding point on the graph of $g(x)$ is $(3,24)$ . What is a possible formula for $g(x)$ in terms of $f(x)$ ?

The following figure shows the graph of $f(x)$ and the graphs of several multiples of $f(x)$ . Suppose that $A$ is the graph of $y=a \cdot f(x)$ , $B$ is the graph of $y=b \cdot f(x)$ , $C$ is the graph of $y=c \cdot f(x)$ , $D$ is the graph of $y=d \cdot f(x)$ . Which of the following statements are true?

The amount of money in your bank account is given by $V(t)$ . Which of the following formulas matches the description "the amount of money I will have in my bank account 30 years from now"?

If $f(x)$ is an even function, determine whether $g(x)=13 f(x-2)$ is odd, even, or neither. State "cannot tell" if there is not enough information to determine.

Suppose the range of $f(x)$ is $3 \leq y \leq 14$ . Let $g(x)=5 f(x-4)$ . There exists some $p$ such that $g(p)=4$ .

The average rate of change of $f(x)$ over $1 \leq x \leq 7$ is -3 . Find the average rate of change of $y=4 f(x)+14$ over $1 \leq x \leq 7$ .

One of these graphs shows a transformation of an exponential function, and one is a transformation of a log function. The remaining graph is not a transformation of either an exponential or log function. Which graph is the transformation of a log function?