Question

the parent function is $f(x)=2^{x - 6}+3$ and the new function is $g(x)=2^{x - 3}+2$
how does the graph change from $f(x)$ to $g(x)$?
select each correct answer.

□ the graph is shifted 3 units right.

□ the graph is shifted 1 unit down.

□ the graph is shifted 1 unit up.

□ the graph is shifted 3 units left.

Explanation:

Step1: Analyze horizontal shift

For exponential functions of the form \(y = a^{x - h}+k\), horizontal shift is determined by \(h\). In \(f(x)=2^{x - 6}+3\), \(h_1 = 6\); in \(g(x)=2^{x - 3}+2\), \(h_2 = 3\). The change in \(h\) is \(h_2 - h_1=3 - 6=- 3\)? Wait, no, the formula for horizontal shift: if we have \(y = a^{x - h}\), shifting left by \(c\) units is \(y = a^{x-(h - c)}=a^{x - h + c}\). So from \(x - 6\) to \(x - 3\), we can write \(x - 3=(x - 6)+3\), so this is a shift of 3 units to the left? Wait, no: Let's let \(u=x - 6\), then \(f(x)=2^{u}+3\). For \(g(x)=2^{x - 3}+2\), \(x - 3=(x - 6)+3\), so \(u'=u + 3\), which means \(x\) increases by 3 for the same \(u'\), so the graph shifts left by 3 units? Wait, no, horizontal shift: if \(y = f(x - h)\), shifting left by \(c\) is \(y = f(x-(h - c))=f(x - h + c)\). So from \(h = 6\) to \(h = 3\), the change is \(h\) decreases by 3, so the graph shifts left by 3 units? Wait, let's take a point. For \(f(x)\), when \(x = 6\), \(f(6)=2^{0}+3 = 4\). For \(g(x)\), when \(x = 3\), \(g(3)=2^{0}+2 = 3\). Wait, the point \((6,4)\) in \(f(x)\) corresponds to \((3,3)\) in \(g(x)\)? No, wait, maybe I messed up. Let's rewrite \(g(x)\) as \(2^{(x - 6)+3}+2\). So \(g(x)=2^{x - 6}\times2^{3}+2=8\times2^{x - 6}+2\). But that's not helpful. Wait, the standard form is \(y = a^{x - h}+k\), where horizontal shift is \(h\) (right if \(h>0\), left if \(h<0\) relative to \(y = a^{x}\)). So \(f(x)=2^{x - 6}+3\) has horizontal shift 6 units right, \(g(x)=2^{x - 3}+2\) has horizontal shift 3 units right. So the change in horizontal shift is from 6 right to 3 right, which is a shift of 3 units left (because 6 - 3 = 3, so moving left by 3 units).

Step2: Analyze vertical shift

The vertical shift is determined by \(k\). In \(f(x)\), \(k_1 = 3\); in \(g(x)\), \(k_2 = 2\). The change is \(k_2 - k_1=2 - 3=-1\), so the graph shifts down by 1 unit.

So the correct options are "The graph is shifted 1 unit down" and "The graph is shifted 3 units left".

Answer:

The graph is shifted 1 unit down, The graph is shifted 3 units left (corresponding to the options: "The graph is shifted 1 unit down" and "The graph is shifted 3 units left")