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 5 0 $-2$

Explanation:

Step1: Recall horizontal translation rule

For a function \( y = f(x) \), a horizontal translation left by \( h \) units gives \( y = f(x + h) \). But here \( g(x)=(2.5)^{x - h} \) and \( f(x)=(2.5)^x \). So \( g(x)=f(x - h) \), and horizontal translation: if \( g(x) \) is left translation of \( f(x) \), then \( g(x)=f(x + |h|) \), so \( x - h=x + |h| \), which means \( -h = |h| \) (since left translation). Wait, alternatively, let's find a point on \( f(x) \) and \( g(x) \).

Step2: Find a point on \( f(x) \)

For \( f(x)=(2.5)^x \), when \( x = 0 \), \( f(0)=(2.5)^0 = 1 \). So the point \( (0,1) \) is on \( f(x) \).

Step3: Find the corresponding point on \( g(x) \)

Looking at the graph, the red graph \( g(x) \) passes through \( (-2,1) \)? Wait, no. Wait, blue is \( f(x) \), red is \( g(x) \). Wait, blue \( f(x) \) has a point at \( (0,1) \), and red \( g(x) \) has a point at \( (-2,1) \)? Wait, no, looking at the grid, the blue curve \( f(x) \) passes through \( (0,1) \), and the red curve \( g(x) \) passes through \( (-2,1) \)? Wait, no, let's check the x - coordinates. Wait, maybe another point. Wait, when \( x = 0 \), \( f(0)=1 \), and \( g(x) \) at \( x=-2 \) is 1? Wait, no, let's see the horizontal shift. The blue curve \( f(x)=(2.5)^x \) and red curve \( g(x)=(2.5)^{x - h} \). Let's take the y - intercept. For \( f(x) \), y - intercept is at \( (0,1) \). For \( g(x) \), let's find its y - intercept. Wait, when \( x = 0 \), \( g(0)=(2.5)^{-h} \). But from the graph, \( g(x) \) at \( x=-2 \) is equal to \( f(x) \) at \( x = 0 \)? Wait, no, let's see the shift. If \( f(x) \) is shifted left by \( h \) units to get \( g(x) \), then \( g(x)=f(x + h) \). But the formula given is \( g(x)=(2.5)^{x - h} \), so \( (2.5)^{x - h}=(2.5)^{x + h'} \) where \( h' \) is the left shift. So \( x - h=x + h' \implies -h = h' \). So \( h'=-h \), and since it's left shift, \( h' \) is positive, so \( -h>0 \implies h<0 \)? Wait, maybe better to use points. Let's take the point where \( f(x) \) is at \( (0,1) \), and \( g(x) \) is at \( (-2,1) \)? Wait, no, looking at the graph, the blue curve (f(x)) has a point at (0,1), and the red curve (g(x)) has a point at (-2,1)? Wait, no, the red curve at x=-2 is 1? Wait, no, let's check the x - axis. Wait, the blue curve (f(x)) passes through (0,1), and the red curve (g(x)) passes through (-2,1)? Wait, no, maybe the shift is 2 units left. So if \( f(x) \) is shifted left by 2 units, then \( g(x)=f(x + 2)=(2.5)^{x + 2} \). But the given \( g(x)=(2.5)^{x - h} \), so \( x - h=x + 2 \implies -h = 2 \implies h=-2 \)? Wait, no, wait the problem says "horizontally translated left by a value of h to get the function \( g(x)=(2.5)^{x - h} \)". Wait, the standard horizontal translation: \( y = f(x) \) shifted left by \( k \) units is \( y = f(x + k) \). So here \( g(x)=f(x + h) \) (since left by h), and \( f(x)=(2.5)^x \), so \( g(x)=(2.5)^{x + h} \). But the problem says \( g(x)=(2.5)^{x - h} \). So \( (2.5)^{x + h}=(2.5)^{x - h} \)? No, that can't be. Wait, maybe the problem has a typo, but looking at the options, and the graph, the shift from \( f(x) \) (blue) to \( g(x) \) (red) is 2 units left. So \( g(x)=f(x + 2)=(2.5)^{x + 2} \). But the given \( g(x)=(2.5)^{x - h} \), so \( x - h=x + 2 \implies -h = 2 \implies h=-2 \)? Wait, no, maybe I messed up. Wait, let's take a point on \( f(x) \): when \( x = 0 \), \( f(0)=1 \). On \( g(x) \), when does \( g(x)=1 \)? \( (2.5)^{x - h}=1 \implies x - h = 0 \implies x = h \). So the point \( (h,1) \) is on \( g(x) \). For \( f(x) \), the point \( (0,1) \) is on it. So the shift from \( (0,1) \) to \( (h,1) \) is a left shift, so \( h = 0 - 2=-2 \)? Wait, if it's left shift by 2, then the x - coordinate of the point (0,1) on f(x) moves to x=-2 on g(x). So \( h \) is such that \( x - h=-2 \) when \( x = 0 \) (since \( g(0)=1 \))? Wait, \( g(0)=(2.5)^{-h}=1 \implies -h = 0 \implies h = 0 \)? No, that's not right. Wait, I think I made a mistake. Let's re - express:

The function \( f(x)=(2.5)^x \), after horizontal translation left by \( h \) units, the new function is \( g(x)=f(x + h)=(2.5)^{x + h} \). But the problem states \( g(x)=(2.5)^{x - h} \). So \( (2.5)^{x + h}=(2.5)^{x - h} \) implies \( x + h=x - h \implies 2h = 0 \implies h = 0 \), which is wrong. So there must be a mistake in the problem's formula, but looking at the graph, the shift is 2 units left. So if we consider that the correct formula for left shift is \( g(x)=f(x + h) \), and the given formula is \( g(x)=(2.5)^{x - h} \), then \( x - h=x + h_{shift} \), so \( -h = h_{shift} \). Since \( h_{shift}=2 \) (left shift by 2), then \( -h = 2 \implies h=-2 \). But the options have - 2 as an option. Wait, the options are 2,5,0,-2. So the value of \( h \) is - 2? Wait, no, wait the problem says "horizontally translated left by a value of h to get the function \( g(x)=(2.5)^{x - h} \)". Wait, maybe the formula is \( g(x)=(2.5)^{x + h} \) (left shift), but it's written as \( x - h \), so \( x - h=x + |h| \) when left shift, so \( -h = |h| \), which means \( h \) is negative. So if left shift by 2, then \( h=-2 \).

Answer:

\( \boldsymbol{-2} \)