Question

match each exponential function to its graph. u(x) = 3.5(2)^x, r(x) = -3.5(2)^x, p(x) = 3.5(0.2)^x, n(x) = -3.5(0.2)^x (and four graphs below these functions)

Explanation:

Step1: Analyze \( u(x) = 3.5(2)^x \)

• The base \( 2>1 \), so it's an exponential growth function. When \( x = 0 \), \( u(0)=3.5(2)^0 = 3.5 \). As \( x \) increases, \( y \) increases rapidly. The graph in the top - right (second graph) has a \( y \) - intercept around 3.5 and shows exponential growth, so \( u(x) \) matches the top - right graph.

Step2: Analyze \( r(x)=- 3.5(2)^x \)

• The base \( 2 > 1 \), but there is a negative sign, so it's a reflection of the growth function over the \( x \) - axis. When \( x = 0 \), \( r(0)=-3.5(2)^0=-3.5 \). As \( x \) increases, \( y \) decreases rapidly (since the negative sign flips the growth). The graph in the bottom - right (fourth graph) has a \( y \) - intercept around - 3.5 and shows a decreasing exponential curve, so \( r(x) \) matches the bottom - right graph.

Step3: Analyze \( p(x)=3.5(0.2)^x \)

• The base \( 0.2<1 \), so it's an exponential decay function. When \( x = 0 \), \( p(0)=3.5(0.2)^0 = 3.5 \). As \( x \) increases, \( y \) approaches 0. The graph in the bottom - left (third graph) has a \( y \) - intercept around 3.5 and shows exponential decay, so \( p(x) \) matches the bottom - left graph.

Step4: Analyze \( n(x)=-3.5(0.2)^x \)

• The base \( 0.2 < 1 \), and there is a negative sign, so it's a reflection of the decay function over the \( x \) - axis. When \( x = 0 \), \( n(0)=-3.5(0.2)^0=-3.5 \). As \( x \) increases, \( y \) approaches 0 from the negative side. The graph in the top - left (first graph) has a \( y \) - intercept around - 3.5 and shows a curve that approaches the \( x \) - axis from below, so \( n(x) \) matches the top - left graph.

Answer:

• \( u(x)=3.5(2)^x \): Top - Right Graph
• \( r(x)=-3.5(2)^x \): Bottom - Right Graph
• \( p(x)=3.5(0.2)^x \): Bottom - Left Graph
• \( n(x)=-3.5(0.2)^x \): Top - Left Graph