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 $h$? $circ$ $-2$ $circ$ $0$ $circ$ $2$ $circ$ $5$

Explanation:

Step1: Recall horizontal translation rule

For a function $f(x)$, a horizontal translation left by $h$ units gives $f(x+h)$.

Step2: Match to given $g(x)$

We know $g(x) = (2.5)^{x-h}$, and it equals $f(x+h)=(2.5)^{x+h}$. Set exponents equal:
$x-h = x+h$

Step3: Solve for $h$

Simplify the equation:
$-h = h$
$2h = 0$? No, use graph point: $f(0)=(2.5)^0=1$ (blue at $(0,1)$). $g(x)$ has $(1,1)$? No, $g(-2)=1$: $(2.5)^{-2-h}=1$. Since $a^0=1$, $-2-h=0$ → $h=-2$.

Answer:

-2