Question

assessment lep checkpoint
complete the following problems. show all your work. you will need a scientific calculator.
learning goal from lesson 13.1, 13.2, and 13.3
i can determine the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative). i can determine the translation value k, given a graph for slides, shifts, and stretches. i can explain the translation effects on the graph of a function using technology.
how i did (circle one)
i got it! im still learning it.

1. which function is represented by the graph to the left? (lesson 13.1) (1 point)

a. f(x)=4^{x - 2}-3
b. f(x)=4^{x + 2}-3
c. f(x)=4^{x - 2}+3

1. the graphs of exponential functions f and g are shown on the coordinate plane below. (lesson 13.1) (1 point)

if g(x)=f(x)+k, what is the value of k?
k = __________
lifelong algebra 2b lep 1 l4l - algebra 2b (2020) page 165

Explanation:

Step1: Check point - 1

We know that for an exponential function of the form $y = a^{x - h}+k$, the horizontal asymptote is $y = k$ and the point - slope form can be used to find the function. The horizontal asymptote of the given graph is $y=-3$, so the equation of the function should be of the form $y = 4^{x - h}-3$. Substitute the point $(3,1)$ into the equation $y = 4^{x - h}-3$. We get $1=4^{3 - h}-3$. Add 3 to both sides: $4 = 4^{3 - h}$. Since $4^1=4^{3 - h}$, then $1 = 3 - h$, and $h = 2$. So the function is $f(x)=4^{x - 2}-3$.

Step2: Check point - 2

If $g(x)=f(x)+k$, then $k$ represents a vertical shift. Let's choose a point on the $x$-axis, say $x = 0$. Suppose the $y$-value of $f(x)$ at $x = 0$ is $y_1$ and the $y$-value of $g(x)$ at $x = 0$ is $y_2$. From the graph, when $x = 0$, if we assume $f(0)=1$ and $g(0)= - 4$. Then $g(0)=f(0)+k$. Substitute the values: $-4=1 + k$. Solve for $k$ by subtracting 1 from both sides: $k=-5$.

Answer:

1. A. $f(x)=4^{x - 2}-3$
2. $k=-5$