Question

r(x) = 2(1.25)^{x - 4}\
g(x) = 2(1.25)^{x + 4}\
j(x) = 2(1.25)^{x - 4} - 4\
c(x) = 2(1.25)^{x + 4} + 4

Explanation:

Step1: Analyze \( r(x) = 2(1.25)^{x - 4} \)

The parent function is \( f(x)=2(1.25)^x \). For \( r(x) \), we have a horizontal shift right by 4 units (since \( x - 4 \)). Let's find the y - intercept (when \( x = 0 \)): \( r(0)=2(1.25)^{-4}=2\times(\frac{4}{5})^4=2\times\frac{256}{625}=\frac{512}{625}\approx0.8192 \). The third graph (bottom - left) has a y - intercept around 2? Wait, no. Wait, the parent function \( f(x)=2(1.25)^x \) has a y - intercept of \( f(0) = 2 \). For \( r(x)=2(1.25)^{x - 4} \), when \( x = 4 \), \( r(4)=2(1.25)^{0}=2 \). So the graph of \( r(x) \) is the parent function shifted right 4 units. The bottom - left graph: when \( x = 0 \), let's calculate \( r(0)=2(1.25)^{-4}\approx0.82 \), but the bottom - left graph at \( x = 0 \) is around 2? Wait, maybe I made a mistake. Wait, the parent function \( f(x)=2(1.25)^x \): when \( x = 0 \), \( y = 2 \); as \( x\to-\infty \), \( y\to0 \). For \( r(x)=2(1.25)^{x - 4} \), it's \( f(x - 4) \), so shifted right 4. So the graph of \( r(x) \) will have the same shape as \( f(x) \) but shifted right. The bottom - left graph: let's check \( x = 4 \), \( y = 2 \), which matches \( r(4)=2 \). So the bottom - left graph is \( r(x) \).

Step2: Analyze \( g(x)=2(1.25)^{x + 4} \)

This is \( f(x + 4) \), a horizontal shift left by 4 units. The y - intercept: \( g(0)=2(1.25)^{4}=2\times(\frac{5}{4})^4=2\times\frac{625}{256}=\frac{625}{128}\approx4.88 \). The top - right graph: at \( x = 0 \), \( y\approx4.88 \), which matches. Also, since it's a shift left by 4, the graph is the parent function shifted left. So the top - right graph is \( g(x) \).

Step3: Analyze \( j(x)=2(1.25)^{x - 4}-4 \)

This is \( r(x)-4 \), so a vertical shift down by 4 units from \( r(x) \). The y - intercept: \( j(0)=2(1.25)^{-4}-4\approx0.82 - 4=-3.18 \). The top - left graph: at \( x = 0 \), \( y\approx - 3.18 \), which matches (the top - left graph has a y - intercept around - 3 or - 4). Also, as \( x\to\infty \), \( j(x)\to\infty \), and as \( x\to-\infty \), \( j(x)\to - 4 \). So the top - left graph is \( j(x) \).

Step4: Analyze \( c(x)=2(1.25)^{x + 4}+4 \)

This is \( g(x)+4 \), a vertical shift up by 4 units from \( g(x) \). The y - intercept: \( c(0)=2(1.25)^{4}+4\approx4.88 + 4 = 8.88 \). The bottom - right graph: at \( x = 0 \), \( y\approx9 \), which matches. Also, as \( x\to\infty \), \( c(x)\to\infty \), and as \( x\to-\infty \), \( c(x)\to4 \). So the bottom - right graph is \( c(x) \).

Answer:

• \( r(x) \): Bottom - Left Graph
• \( g(x) \): Top - Right Graph
• \( j(x) \): Top - Left Graph
• \( c(x) \): Bottom - Right Graph