Question

the graph of $y = 2^x$ is shown below.

which of the following is the graph of $y = -1 \cdot 2^{x+3} + 4$
choose 1 answer:
a
b
c
d

Explanation:

Step1: Rewrite the function

First, rewrite the given function to identify transformations:
$$y = -2^{x+3} + 4$$
This is a transformed version of $y=2^x$.

Step2: Identify horizontal shift

Shift $y=2^x$ left 3 units:
$$y=2^{x+3}$$

Step3: Reflect over x-axis

Reflect the graph across the x-axis:
$$y=-2^{x+3}$$

Step4: Vertical shift up

Shift the reflected graph up 4 units:
$$y=-2^{x+3}+4$$

Step5: Analyze key features

• Horizontal asymptote: As $x\to\infty$, $2^{x+3}\to\infty$, so $y\to -\infty$. As $x\to-\infty$, $2^{x+3}\to0$, so $y\to4$. The asymptote is $y=4$.
• y-intercept: When $x=0$, $y=-2^{0+3}+4=-8+4=-4$.
• Behavior: Decreasing (due to the negative coefficient) with asymptote at $y=4$, passing through $(0,-4)$.

Answer:

A. <The graph with horizontal asymptote $y=4$, decreasing, passing through $(0,-4)$>