Multiple Choice
Determine if the function is an exponential function.
If so, identify the power & base, then evaluate for .
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6. Exponential and Logarithmic Functions
Introduction to Exponential Functions
Multiple Choice
Which of the following graphs represents an exponential function?
A
A downward-opening parabola with the equation
B
A straight line with the equation
C
A horizontal line with the equation
D
A graph that passes through the point and increases rapidly as increases, with the equation
Verified step by step guidance1
Recall that an exponential function has the form \(y = a^{x}\), where the variable \(x\) is in the exponent, and \(a\) is a positive constant not equal to 1.
Examine each given equation to identify which one fits the exponential form:
- \(y = -x^{2}\) is a quadratic function because the variable \(x\) is raised to a power, not in the exponent.
- \(y = 2x + 1\) is a linear function because \(x\) is to the first power and not in the exponent.
- \(y = 3\) is a constant function, which is a horizontal line.
Look at the description of the correct graph: it passes through the point \((0,1)\) and increases rapidly as \(x\) increases. This matches the behavior of an exponential function because \(a^{0} = 1\) for any positive \(a\).
Identify the equation \(y = 2^{x}\) as the exponential function since the variable \(x\) is in the exponent and the base 2 is a positive constant greater than 1, which causes the rapid increase.
Conclude that the graph representing the exponential function is the one with the equation \(y = 2^{x}\), passing through \((0,1)\) and increasing rapidly as \(x\) increases.
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