Question

u(x) = 3(2.5)^x + 3; t(x) = 2.5^x - 3; h(x) = -3(3.5)^x; g(x) = -(3.5)^x

Explanation:

Step1: Analyze \( u(x) = 3(2.5)^x + 3 \)

Exponential function \( a^x \) with \( a>1 \) is increasing. Here, \( 2.5>1 \), and the function has a vertical stretch by 3 and vertical shift up 3. As \( x\to-\infty \), \( (2.5)^x\to0 \), so \( u(x)\to 3(0)+3 = 3 \). As \( x\to\infty \), \( (2.5)^x\to\infty \), so \( u(x)\to\infty \). The top - left graph has a horizontal asymptote around \( y = 3 \) (since as \( x\to-\infty \), it approaches \( y = 4\) - close to 3) and increases, so it matches \( u(x) \).

Step2: Analyze \( t(x)=2.5^x - 3 \)

For \( t(x)=2.5^x - 3 \), \( a = 2.5>1 \), so it's an increasing exponential. As \( x\to-\infty \), \( 2.5^x\to0 \), so \( t(x)\to0 - 3=-3 \). As \( x\to\infty \), \( t(x)\to\infty \). The bottom - left graph has a horizontal asymptote around \( y=-3 \) (as \( x\to-\infty \), it approaches \( y = - 3\)) and increases, so it matches \( t(x) \).

Step3: Analyze \( h(x)=-3(3.5)^x \)

For \( h(x)=-3(3.5)^x \), \( a = 3.5>1 \), so \( (3.5)^x \) is increasing, and the negative sign reflects it over the x - axis, and the 3 is a vertical stretch. As \( x\to-\infty \), \( (3.5)^x\to0 \), so \( h(x)\to0 \). As \( x\to\infty \), \( (3.5)^x\to\infty \), so \( h(x)\to-\infty \). The top - right graph: as \( x\to-\infty \), it approaches \( y = 0\) (horizontal asymptote) and as \( x\to\infty \), it goes to \( -\infty \), and the rate of decrease should be faster than \( g(x) \) (since it has a vertical stretch of 3). So it matches \( h(x) \).

Step4: Analyze \( g(x)=-(3.5)^x \)

For \( g(x)=-(3.5)^x \), \( a = 3.5>1 \), reflected over x - axis. As \( x\to-\infty \), \( (3.5)^x\to0 \), so \( g(x)\to0 \). As \( x\to\infty \), \( g(x)\to-\infty \). The bottom - right graph: as \( x\to-\infty \), it approaches \( y = 0\) and as \( x\to\infty \), it goes to \( -\infty \), and the rate of decrease is slower than \( h(x) \) (since no vertical stretch of 3), so it matches \( g(x) \).

Answer:

• Top - left graph: \( u(x)=3(2.5)^x + 3 \)
• Top - right graph: \( h(x)=-3(3.5)^x \)
• Bottom - left graph: \( t(x)=2.5^x - 3 \)
• Bottom - right graph: \( g(x)=-(3.5)^x \)