Question

n(x) = -2(1.5)^x - 2.5
r(x) = 2(1.5)^{x + 2.5}
w(x) = 2(1.5)^{x - 2.5}
q(x) = -2(1.5)^x + 2.5

Explanation:

Step1: Analyze \( n(x) = -2(1.5)^x - 2.5 \)

The base of the exponential is \( 1.5>1 \), so \( (1.5)^x \) is an increasing exponential. The negative sign in front of the exponential term reflects it over the x - axis, making it decreasing. The vertical shift is \( - 2.5 \), so the horizontal asymptote is \( y=-2.5 \). Let's check the y - intercept: when \( x = 0 \), \( n(0)=-2(1)-2.5=-4.5 \). The first graph (top - left) has a y - intercept around - 4 and a horizontal asymptote around - 2.5? Wait, no, let's re - check. Wait, when \( x = 0 \), \( n(0)=-2(1)-2.5=-4.5 \). The top - left graph has a y - intercept around - 4? Wait, maybe I made a mistake. Let's check the other functions.

Step2: Analyze \( r(x)=2(1.5)^{x + 2.5} \)

Using the property of exponents \( a^{m + n}=a^m\times a^n \), \( r(x)=2\times(1.5)^{2.5}\times(1.5)^x \). Since \( 1.5>1 \), this is an increasing exponential. The y - intercept when \( x = 0 \) is \( r(0)=2\times(1.5)^{2.5}\approx2\times2.847\approx5.694 \). The bottom - right graph has a y - intercept around 6, which is close.

Step3: Analyze \( w(x)=2(1.5)^{x-2.5} \)

Using the property \( a^{m - n}=\frac{a^m}{a^n} \), \( w(x)=2\times\frac{(1.5)^x}{(1.5)^{2.5}}\approx2\times\frac{(1.5)^x}{2.847}\approx0.702(1.5)^x \). This is an increasing exponential (since \( 1.5>1 \)) with a y - intercept when \( x = 0 \): \( w(0)=2\times(1.5)^{-2.5}=2\times\frac{1}{(1.5)^{2.5}}\approx2\times\frac{1}{2.847}\approx0.702 \). The top - right graph has a y - intercept around 1, which is close.

Step4: Analyze \( q(x)=-2(1.5)^x + 2.5 \)

This is a reflected (over x - axis) exponential (since the coefficient of \( (1.5)^x \) is negative) with a vertical shift of \( + 2.5 \), so the horizontal asymptote is \( y = 2.5 \). When \( x = 0 \), \( q(0)=-2(1)+2.5 = 0.5 \)? Wait, no: \( q(0)=-2(1)+2.5 = 0.5 \)? Wait, no, \( -2 + 2.5 = 0.5 \)? Wait, no, \( -2\times1+2.5 = 0.5 \). The bottom - left graph has a y - intercept at \( y = 0 \) (when \( x = 0 \), the graph crosses the y - axis at 0). Wait, maybe I made a mistake. Let's re - check the bottom - left graph: when \( x = 0 \), the graph passes through (0,0). Let's check \( q(x) \) at \( x = 0 \): \( q(0)=-2(1)+2.5 = 0.5 \), close to 0. The bottom - left graph: when \( x = 0 \), \( y = 0 \)? Wait, no, the bottom - left graph at \( x = 0 \) is at \( y = 0 \)? Wait, maybe the function for the bottom - left is \( q(x) \). Wait, let's list the graphs:

Top - left: Let's assume it's \( n(x) \) (since it's a decreasing exponential with a horizontal asymptote around \( y=-2.5 \) and y - intercept around - 4.5)

Top - right: \( w(x) \) (increasing, y - intercept around 1)

Bottom - left: \( q(x) \) (decreasing, horizontal asymptote \( y = 2.5 \), y - intercept around 0)

Bottom - right: \( r(x) \) (increasing, y - intercept around 6)

Let's confirm \( r(x) \): \( r(x)=2(1.5)^{x + 2.5} \). When \( x = 0 \), \( r(0)=2(1.5)^{2.5}\approx2\times2.847\approx5.69\approx6 \), which matches the bottom - right graph's y - intercept. The bottom - right graph is increasing, which matches \( r(x) \) (since the base is \( 1.5>1 \) and the coefficient is positive).

For \( w(x)=2(1.5)^{x - 2.5} \), when \( x = 0 \), \( w(0)=2(1.5)^{-2.5}\approx2\times0.351\approx0.702 \), and the top - right graph has a y - intercept around 1, which is a reasonable approximation.

For \( q(x)=-2(1.5)^x+2.5 \), when \( x = 0 \), \( q(0)=-2 + 2.5 = 0.5 \), and the bottom - left graph has a y - intercept around 0, which is close (maybe due to graph scaling).

For \( n(x)=-2(1.5)^x-2.5 \), when \( x = 0 \), \( n(0)=-2 - 2.5=-4.5 \), and the top - left graph has a y - intercept around - 4, which is close (scaling again).

Let's match each function to the graph:

- \( n(x)=-2(1.5)^x - 2.5 \): Top - Left Graph (decreasing, y - intercept ~ - 4.5, horizontal asymptote \( y=-2.5 \))
- \( r(x)=2(1.5)^{x + 2.5} \): Bottom - Right Graph (increasing, y - intercept ~ 6)
- \( w(x)=2(1.5)^{x-2.5} \): Top - Right Graph (increasing, y - intercept ~ 1)
- \( q(x)=-2(1.5)^x + 2.5 \): Bottom - Left Graph (decreasing, y - intercept ~ 0.5, horizontal asymptote \( y = 2.5 \))

Answer:

• \( n(x) \): Top - Left Graph
• \( r(x) \): Bottom - Right Graph
• \( w(x) \): Top - Right Graph
• \( q(x) \): Bottom - Left Graph