Question

the graph shows the function $f(x)=(2.5)^x$ was horizontally translated left by a value of $h$ to get the function $g(x)=(2.5)^{x - h}$. what is the value of $h$? -2 0 2 5

Explanation:

Step1: Recall horizontal translation rule

For a function \(y = a^{x - h}\), if \(h>0\), the graph of \(y = a^{x}\) is translated \(h\) units to the right; if \(h < 0\), the graph of \(y=a^{x}\) is translated \(|h|\) units to the left.

Step2: Use a point - by - point comparison

The original function \(f(x)=(2.5)^{x}\) passes through the point \((0,1)\) (since \(f(0)=(2.5)^{0}=1\)).
The translated function \(g(x)=(2.5)^{x - h}\) passes through the point \((- 2,1)\).
Substitute \(x=-2\) and \(g(-2) = 1\) into \(g(x)=(2.5)^{x - h}\):

$$1=(2.5)^{-2 - h}$$

Since \(a^{0}=1\) for \(a>0,a
eq1\), we have \(-2 - h=0\).

Step3: Solve for \(h\)

$$h=-2$$

Answer:

\(-2\)