Here's an exponential function. y=((1)/(4))^x (a) Choose the graph that best represents the shape of this function. square

To determine the graph that best represents the shape of the exponential function \( y = \left(\frac{1}{4}\right)^x \), we need to understand the behavior of this function.<br /><br />1. **Base Analysis**: The base of the exponential function is \(\frac{1}{4}\), which is a fraction less than 1. This means the function is decreasing as \( x \) increases.<br /><br />2. **Behavior for Positive and Negative \( x \)**:<br /> - For \( x > 0 \): Since \(\left(\frac{1}{4}\right)^x\) is a fraction raised to a positive power, it gets smaller and smaller, approaching 0 but never reaching it.<br /> - For \( x < 0 \): \(\left(\frac{1}{4}\right)^x\) becomes \(\left(\frac{4}{1}\right)^{-x} = 4^x\), which grows exponentially as \( x \) becomes more negative.<br /><br />3. **Asymptote**: The horizontal asymptote for this function is \( y = 0 \). As \( x \) increases, \( y \) approaches 0 from above.<br /><br />4. **Key Points**:<br /> - When \( x = 0 \): \( y = \left(\frac{1}{4}\right)^0 = 1 \)<br /> - When \( x = 1 \): \( y = \left(\frac{1}{4}\right)^1 = \frac{1}{4} \)<br /> - When \( x = -1 \): \( y = \left(\frac{1}{4}\right)^{-1} = 4 \)<br /><br />Given these characteristics, the graph of \( y = \left(\frac{1}{4}\right)^x \) will:<br />- Start at \( (0, 1) \)<br />- Decrease rapidly as \( x \) increases<br />- Approach the x-axis (y = 0) as \( x \) increases<br />- Increase rapidly as \( x \) decreases<br /><br />The correct graph will show a curve that starts high on the left side (for negative \( x \)), passes through \( (0, 1) \), and then decreases towards the x-axis as \( x \) increases.<br /><br />Without seeing the actual graphs, I can describe what the correct graph should look like:<br />- It should be a smooth curve starting from the top left, passing through the point \( (0, 1) \), and then descending towards the x-axis as it moves to the right.<br /><br />If you have multiple-choice graphs to choose from, select the one that matches this description.