Graph the equation on​ paper, and then choose the correct graph.

Question

yequals6 Superscript x
Question content area bottom left
Part 1
Choose the correct graph on the right.
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Question content area right
Part 1
A.
-5
5
-5
5
x
y

A coordinate system has a horizontal axis labeled from negative 5 to 5 in increments of 1 and a vertical axis labeled from negative 5 to 5 in increments of 1. A line rises from left to right passing through the points (0, 1) and (1, 7).
B.
-5
5
-5
5
x
y

A coordinate system has a horizontal axis labeled from negative 5 to 5 in increments of 1 and a vertical axis labeled from negative 5 to 5 in increments of 1. From left to right, a curve rises steeply in quadrant 4, crosses the horizontal axis at 1, and then rises at a decreasing rate in quadrant 1 through approximately (4, 0.8). As the horizontal values decrease towards 0, the graph approaches the negative vertical axis from the right.
C.
-5
5
-5
5
x
y

A coordinate system has a horizontal axis labeled from negative 5 to 5 in increments of 1 and a vertical axis labeled from negative 5 to 5 in increments of 1. A curve falls steeply from left to right, passing through the point left parenthesis negative 1 comma 2 right parenthesis, then reaches a minimum at (0, 1), and then rises steeply from left to right passing through the point left parenthesis 1 comma 2 right parenthesis.
D.
-5
5
-5
5
x
y

Answer 1

To graph the equation \( y = 6^x \) and choose the correct graph, let's analyze the function:

1. Behavior of the Function:

   • When \( x = 0 \): \( y = 6^0 = 1 \).
   • When \( x = 1 \): \( y = 6^1 = 6 \).
   • When \( x = -1 \): \( y = 6^{-1} = \frac{1}{6} \approx 0.17 \).
   • When \( x = -2 \): \( y = 6^{-2} = \frac{1}{36} \approx 0.027 \).

2. Key Features:

   • The graph will always be above the x-axis for all real numbers \( x \) (since exponential functions do not touch the x-axis).
   • As \( x \) increases, \( y \) will increase rapidly.
   • As \( x \) decreases (moving left), \( y \) approaches 0 but never reaches it, i.e., the graph approaches the x-axis asymptotically.

With this analysis in mind, let's look at the options:

• Option A: Describes a line rising from left to right, which does not match the exponential function.

• Option B: Describes a curve that rises steeply and approaches the negative vertical axis from the right. This matches the behavior of \( y = 6^x \) (it rises steeply, crosses the y-axis at (0, 1), and approaches the x-axis as \( x \) decreases).

• Option C: Describes a curve falling steeply then rising again, which does not represent the exponential function.

• Option D: No information is provided, so we cannot evaluate it.

Based on the characteristics of the function \( y = 6^x \), the correct graph is B.