\( \begin{array}{l} \text { My IXL }\\ \text { Learning }\\ \text { Assessment }\\ \text { Analytics }\\ 5 \text { Match exponential functions and graphs PCX }\\ \text { Match each exponential function to its graph. }\\ m(x)=2(3)^{x}\\ q(x)=2\left(\frac{1}{3}\right)^{x}\\ s(x)=3^{x}+1\\ w(x)=\left(\frac{1}{3}\right)^{x}+1 \end{array} \)

To match each exponential function to its corresponding graph, we need to analyze the characteristics of each function. Let's break down the functions one by one: 1. **Function \( m(x) = 2(3)^x \)**: - This is an exponential growth function because the base \( 3 \) is greater than \( 1 \). - The graph will start from \( (0, 2) \) since \( m(0) = 2(3)^0 = 2 \). - As \( x \) increases, \( m(x) \) will increase rapidly. 2. **Function \( q(x) = 2\left(\frac{1}{3}\right)^x \)**: - This is an exponential decay function because the base \( \frac{1}{3} \) is less than \( 1 \). - The graph will also start from \( (0, 2) \) since \( q(0) = 2\left(\frac{1}{3}\right)^0 = 2 \). - As \( x \) increases, \( q(x) \) will decrease towards \( 0 \). 3. **Function \( s(x) = 3^x + 1 \)**: - This is also an exponential growth function with a vertical shift upwards by \( 1 \). - The graph will start from \( (0, 4) \) since \( s(0) = 3^0 + 1 = 4 \). - As \( x \) increases, \( s(x) \) will increase rapidly, but it will never reach \( 1 \) (the horizontal asymptote). 4. **Function \( w(x) = \left(\frac{1}{3}\right)^x + 1 \)**: - This is an exponential decay function with a vertical shift upwards by \( 1 \). - The graph will start from \( (0, 2) \) since \( w(0) = \left(\frac{1}{3}\right)^0 + 1 = 2 \). - As \( x \) increases, \( w(x) \) will decrease towards \( 1 \) (the horizontal asymptote). ### Summary of Characteristics: - **Growth Functions**: - \( m(x) = 2(3)^x \): Starts at \( (0, 2) \) and increases rapidly. - \( s(x) = 3^x + 1 \): Starts at \( (0, 4) \) and increases rapidly. - **Decay Functions**: - \( q(x) = 2\left(\frac{1}{3}\right)^x \): Starts at \( (0, 2) \) and decreases towards \( 0 \). - \( w(x) = \left(\frac{1}{3}\right)^x + 1 \): Starts at \( (0, 2) \) and decreases towards \( 1 \). ### Matching Functions to Graphs: - **Graph of \( m(x) \)**: Exponential growth starting from \( (0, 2) \). - **Graph of \( q(x) \)**: Exponential decay starting from \( (0, 2) \) and decreasing towards \( 0 \). - **Graph of \( s(x) \)**: Exponential growth starting from \( (0, 4) \). - **Graph of \( w(x) \)**: Exponential decay starting from \( (0, 2) \) and decreasing towards \( 1 \). Now, you can match the functions to their respective graphs based on these characteristics.