Question

1. graph $y=2^x - 5$. how does this graph compare to that of $y=2^x$?

Explanation:

Step1: Identify parent function

The parent function is $y=2^x$, which is an exponential growth function with a horizontal asymptote at $y=0$, passing through the point $(0,1)$.

Step2: Analyze transformation rule

For a function $y=f(x)-k$, the graph of $y=f(x)$ is shifted downward by $k$ units. Here, $f(x)=2^x$ and $k=5$.

Step3: Determine key features of transformed graph

• Horizontal asymptote: Shifts from $y=0$ to $y=-5$
• y-intercept: For $y=2^x-5$, when $x=0$, $y=2^0-5=1-5=-4$, so it passes through $(0,-4)$
• All points on $y=2^x$ are moved down 5 units.

Answer:

The graph of $y=2^x - 5$ is the graph of $y=2^x$ shifted vertically downward by 5 units. Specifically:

• The horizontal asymptote of $y=2^x$ (which is $y=0$) moves to $y=-5$ for $y=2^x - 5$.
• Every point $(x, y)$ on $y=2^x$ corresponds to a point $(x, y-5)$ on $y=2^x - 5$.
• The y-intercept changes from $(0, 1)$ (for $y=2^x$) to $(0, -4)$ (for $y=2^x - 5$).