\( \leftarrow \begin{array}{l}\text { Graph the equation on paper, and then choose the correct } \\ \text { graph on the right. } \\ y=\left(\frac{1}{5}\right)^{x}\end{array} \).

To graph the equation \( y = \left(\frac{1}{5}\right)^{x} \), we can analyze its characteristics: 1. **Base**: The base \( \frac{1}{5} \) is a fraction less than 1, which means the function will be a decreasing exponential function. 2. **Intercept**: When \( x = 0 \): \[ y = \left(\frac{1}{5}\right)^{0} = 1 \] So, the graph will pass through the point (0, 1). 3. **Behavior as \( x \) increases**: As \( x \) increases, \( y \) will decrease towards 0 but never actually reach it. For example: - When \( x = 1 \): \[ y = \left(\frac{1}{5}\right)^{1} = \frac{1}{5} = 0.2 \] - When \( x = 2 \): \[ y = \left(\frac{1}{5}\right)^{2} = \frac{1}{25} = 0.04 \] 4. **Behavior as \( x \) decreases**: As \( x \) becomes more negative, \( y \) will increase. For example: - When \( x = -1 \): \[ y = \left(\frac{1}{5}\right)^{-1} = 5 \] - When \( x = -2 \): \[ y = \left(\frac{1}{5}\right)^{-2} = 25 \] 5. **Asymptote**: The x-axis (y = 0) is a horizontal asymptote, meaning the graph approaches this line but never touches it. ### Summary of the Graph: - The graph starts high on the left (for negative \( x \)), crosses the y-axis at (0, 1), and decreases towards the x-axis as \( x \) increases. - The graph is always positive and approaches 0 as \( x \) goes to infinity. ### Choosing the Correct Graph: When looking at the options provided, you should look for a graph that: - Starts high on the left side, - Passes through (0, 1), - Decreases towards the x-axis without touching it. If you have specific graph options to choose from, you can compare them to the characteristics described above to find the correct one.