Question

what you notice about the domain, range, intercepts, and asymptote of $y=2^x + 3$.domain:range:asymptote:9. graph $y=2^x - 5$. how does this graph compare to that of $y=2^x$?based on what you know about transformations of functions, describe in words how $y=2^x - 5$ transforms the graph of the parent function $y=2^x$.$y= 2^x -5$ transforms the graph by moving 5 downdiscuss what you notice about the domain, range, intercepts, and asymptote of $y=2^x - 5$.

Explanation:

Step1: Find domain of $y=2^x -5$

Exponential functions accept all real $x$.
Domain: $(-\infty, \infty)$

Step2: Find range of $y=2^x -5$

For $y=2^x$, range is $(0, \infty)$. Subtract 5:
Range: $(-5, \infty)$

Step3: Find x-intercept

Set $y=0$, solve for $x$:
$0=2^x -5 \implies 2^x=5 \implies x=\log_2 5$

Step4: Find y-intercept

Set $x=0$, solve for $y$:
$y=2^0 -5=1-5=-4$

Step5: Find horizontal asymptote

As $x\to-\infty$, $2^x\to0$, so $y\to0-5=-5$.
Asymptote: $y=-5$

Answer:

• Domain: All real numbers, or $(-\infty, \infty)$
• Range: All real numbers greater than -5, or $(-5, \infty)$
• Intercepts: x-intercept at $(\log_2 5, 0)$, y-intercept at $(0, -4)$
• Asymptote: Horizontal asymptote at $y=-5$