Question

t(x) = 2(1.5)^{x + 4}\
z(x) = -3(1.5)^x\
c(x) = -2(1.5)^x + 4\
g(x) = 3(1.5)^x - 4

Explanation:

Step1: Analyze \( t(x) = 2(1.5)^{x + 4} \)

Rewrite \( t(x) \) using exponent rules: \( t(x)=2(1.5)^4(1.5)^x \). The base \( 1.5>1 \), so it's an exponential growth. When \( x = - 4 \), \( t(-4)=2(1.5)^0 = 2 \)? Wait, no, \( x+4 = 0\) when \( x=-4 \), so \( t(-4)=2(1.5)^0=2 \)? Wait, actually, let's check the y - intercept. When \( x = 0 \), \( t(0)=2(1.5)^4\approx2\times5.0625 = 10.125\), close to 10. The first graph (top - left) has a y - intercept near 10 and is increasing (since growth), so \( t(x) \) matches the top - left graph.

Step2: Analyze \( z(x)=-3(1.5)^x \)

The base \( 1.5 > 1 \), and the coefficient is negative, so it's an exponential decay (reflected over x - axis) and decreasing as \( x \) increases. When \( x = 0 \), \( z(0)=-3(1.5)^0=-3 \). The bottom - left graph has a y - intercept near - 3 and is decreasing, so \( z(x) \) matches the bottom - left graph.

Step3: Analyze \( c(x)=-2(1.5)^x + 4 \)

The base \( 1.5>1 \), coefficient of exponential term is negative (so reflection over x - axis) and then shifted up by 4. When \( x = 0 \), \( c(0)=-2(1.5)^0 + 4=-2 + 4 = 2 \). The bottom - right graph has a y - intercept of 2 and is decreasing (since the exponential term with negative coefficient), so \( c(x) \) matches the bottom - right graph.

Step4: Analyze \( g(x)=3(1.5)^x-4 \)

The base \( 1.5 > 1 \), so exponential growth, shifted down by 4. When \( x = 0 \), \( g(0)=3(1.5)^0-4=3 - 4=-1\)? Wait, no, \( 3(1)^0-4=-1 \)? Wait, no, \( 3(1.5)^0=3\), so \( g(0)=3 - 4=-1 \)? Wait, the top - right graph has a y - intercept near 0 (close to - 1? Wait, the top - right graph at \( x = 0 \) is around \( y = 0\)? Wait, when \( x = 0 \), \( g(0)=3(1)-4=-1\), and as \( x \) increases, \( g(x) \) increases. The top - right graph is increasing and has a y - intercept around 0 (close to - 1), so \( g(x) \) matches the top - right graph.

Answer:

Top - Left Graph: \( t(x)=2(1.5)^{x + 4}\)

Top - Right Graph: \( g(x)=3(1.5)^x-4\)

Bottom - Left Graph: \( z(x)=-3(1.5)^x\)

Bottom - Right Graph: \( c(x)=-2(1.5)^x + 4\)