Question

write a function g(x) that represents the exponential function $f(x)=3^{x}$ after a vertical stretch by a factor of 4 and a reflection across the x-axis. graph both functions.

find the equation for g(x).
$g(x)=\square$

Explanation:

Step1: Apply vertical stretch by 4

A vertical stretch by factor $k$ transforms $f(x)$ to $k\cdot f(x)$. Here, $k=4$, so:
$4\cdot f(x) = 4\cdot 3^x$

Step2: Reflect across x-axis

A reflection across the x-axis multiplies the function by $-1$:
$-1\cdot 4\cdot 3^x = -4\cdot 3^x$

Answer:

$g(x) = -4\cdot 3^x$

Graphing Notes (for context):

• For $f(x)=3^x$: This is a standard exponential growth function, passing through $(0,1)$, increasing as $x$ increases, approaching 0 as $x\to-\infty$.
• For $g(x)=-4\cdot 3^x$: This is a vertically stretched (by 4) and reflected (over x-axis) version, passing through $(0,-4)$, decreasing as $x$ increases, approaching 0 as $x\to-\infty$, and heading to $-\infty$ as $x\to+\infty$.