Graph the exponential function.

Question

y equals 5 Superscript negative x
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Part 1
Choose the correct graph of y equals 5 Superscript negative x on the right.
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Question content area right
Part 1
-5
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A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 25 to 25 in increments of 5. From left to right, a curve falls steeply in quadrant 1, crosses the x-axis at 1, and then falls shallowly in quadrant two passing through (1, 0.2). As x increases towards positive infinity, the curve approaches the positive x-axis from below.
-5
5
-25
25
x
y

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 25 to 25 in increments of 5. From left to right, a curve falls steeply in quadrant 2 passing through (negative 1, 5), crosses the y-axis at 1, and then falls shallowly in quadrant 1 passing through approximately (1, 0.2). As x increases towards positive infinity, the curve approaches the positive x-axis from above.
-5
5
-25
25
x
y

A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 25 to 25 in increments of 5. Upper A curve rises from left to right at an increasing rate passing through the point left parenthesis negative 1 comma negative 1 right parenthesis to the origin comma then rises from left to right at a decreasing rate passing through the point left parenthesis 1 comma 1 right parenthesis. All coordinates are approximate.
-5
5
-25
25
x
y

Answer 1

To graph the exponential function \( y = 5^{-x} \), let's analyze its characteristics.

1. Function Behavior:

   • As \( x \) increases, \( y = 5^{-x} \) decreases because the base \( 5 \) is raised to a negative power.
   • As \( x \) approaches positive infinity, \( y \) approaches \( 0 \) (the horizontal asymptote).
   • As \( x \) approaches negative infinity, \( y \) approaches positive infinity.

2. Key Points:

   • When \( x = 0 \): \( y = 5^0 = 1 \).
   • When \( x = 1 \): \( y = 5^{-1} = \frac{1}{5} = 0.2 \).
   • When \( x = -1 \): \( y = 5^{1} = 5 \).

3. Graph Shape:

   • The graph starts high on the left (as \( x \to -\infty, y \to \infty \)), crosses the y-axis at \( (0, 1) \), decreases quickly, and approaches \( 0 \) without ever touching or crossing the x-axis.

Based on this analysis, the correct description corresponds to a curve that:

• Falls steeply as it moves from the positive y-axis towards negative x-values,
• Crosses the y-axis at \( (0, 1) \),
• Falls shallowly as it approaches the x-axis but never crosses it.

Given your descriptions, the second option appears to be correct:

• The curve falls steeply in quadrant 2, passing through \( (-1, 5) \) and crosses the y-axis at \( (0, 1) \), then falls shallowly in quadrant 1 passing through approximately \( (1, 0.2) \), and as \( x \) increases, it approaches the x-axis from above.

This aligns well with the behavior of the graph for the function \( y = 5^{-x} \).