Question

z(x) = 4(0.75)^{x + 2} + 4
v(x) = -4(0.75)^{x + 2} - 4
s(x) = -4(0.75)^{x - 2} + 4
m(x) = 4(0.75)^{x - 2} - 4

Explanation:

Step1: Analyze \( z(x) = 4(0.75)^{x + 2}+4 \)

The base \( 0.75\in(0,1) \), so it's a decreasing exponential. The vertical shift is \( +4 \), horizontal shift is \( - 2 \) (since \( x+2=x - (- 2) \)). As \( x\to-\infty \), \( (0.75)^{x + 2}\to\infty \), so \( z(x)\to\infty \); as \( x\to\infty \), \( (0.75)^{x + 2}\to0 \), so \( z(x)\to4 \). The first graph (top - left) has a horizontal asymptote around \( y = 4 \) and decreases towards it, so \( z(x) \) matches the top - left graph.

Step2: Analyze \( v(x)=-4(0.75)^{x + 2}-4 \)

The base \( 0.75\in(0,1) \), so \( (0.75)^{x + 2} \) is decreasing, but with a negative coefficient \( - 4 \), it becomes increasing. The vertical shift is \( - 4 \), horizontal shift is \( - 2 \). As \( x\to-\infty \), \( (0.75)^{x + 2}\to\infty \), so \( v(x)\to-\infty \); as \( x\to\infty \), \( (0.75)^{x + 2}\to0 \), so \( v(x)\to - 4 \). The bottom - right graph has a horizontal asymptote around \( y=-4 \) and increases towards it, so \( v(x) \) matches the bottom - right graph.

Step3: Analyze \( s(x)=-4(0.75)^{x - 2}+4 \)

The base \( 0.75\in(0,1) \), \( (0.75)^{x - 2} \) is decreasing, with a negative coefficient \( - 4 \), it becomes increasing. Horizontal shift is \( + 2 \) (since \( x - 2=x-(2) \)), vertical shift is \( + 4 \). As \( x\to-\infty \), \( (0.75)^{x - 2}\to\infty \), so \( s(x)\to-\infty \); as \( x\to\infty \), \( (0.75)^{x - 2}\to0 \), so \( s(x)\to4 \). The top - right graph has a horizontal asymptote around \( y = 4 \) and increases towards it, so \( s(x) \) matches the top - right graph.

Step4: Analyze \( m(x)=4(0.75)^{x - 2}-4 \)

The base \( 0.75\in(0,1) \), so it's a decreasing exponential. Horizontal shift is \( + 2 \), vertical shift is \( - 4 \). As \( x\to-\infty \), \( (0.75)^{x - 2}\to\infty \), so \( m(x)\to\infty \); as \( x\to\infty \), \( (0.75)^{x - 2}\to0 \), so \( m(x)\to - 4 \). The bottom - left graph has a horizontal asymptote around \( y=-4 \) and decreases towards it, so \( m(x) \) matches the bottom - left graph.

Answer:

Top - left graph: \( z(x) = 4(0.75)^{x + 2}+4 \)

Top - right graph: \( s(x)=-4(0.75)^{x - 2}+4 \)

Bottom - left graph: \( m(x)=4(0.75)^{x - 2}-4 \)

Bottom - right graph: \( v(x)=-4(0.75)^{x + 2}-4 \)