Question

starting with the graph of $f(x) = 9^x$, write the equation of the graph that results when:
(a) $f(x)$ is shifted 8 units downwards.
$y = $ enter your answer
(b) $f(x)$ is shifted 8 units to the left.
$y = $ enter your answer
(c) $f(x)$ is reflected about the $x$-axis.
$y = $ enter your answer

Explanation:

Step1: Vertical shift formula

For a function \(y = f(x)\), shifting \(k\) units downwards gives \(y=f(x)-k\). Here \(f(x) = 9^{x}\) and \(k = 8\).
So the equation is \(y=9^{x}-8\).

Step2: Horizontal shift formula

For a function \(y = f(x)\), shifting \(h\) units to the left gives \(y = f(x + h)\). Here \(f(x)=9^{x}\) and \(h = 8\).
So the equation is \(y=9^{x + 8}\).

Step3: Reflection about the \(x\) - axis formula

For a function \(y = f(x)\), reflection about the \(x\) - axis gives \(y=-f(x)\). Here \(f(x)=9^{x}\).
So the equation is \(y=-9^{x}\).

Answer:

(a) \(y = 9^{x}-8\)
(b) \(y = 9^{x + 8}\)
(c) \(y=-9^{x}\)