Chapter 9.1, Problem 74HP

To determine

The function that increases at a faster rate than the others.

Answer to Problem 74HP

The exponential function f (x) eventually exceeds the others.

Explanation of Solution

Given information:

Use tables and graphs to compare and contrast an exponential function f (x) = ab^x + c, where a ≠ 0, b > 0, and b ≠ 1, a quadratic function g(x) = ax² + c, and a linear function h(x) = ax + c. Include intercepts, portions of the graph where the functions are increasing, decreasing, positive, or negative, relative maxima and minima, symmetries, and end behavior

Formula used:

g(x) has a minimum at (0, 1).

Calculation:

Use basic values for each variable. Sample answer: Suppose a = 1, b = 2, and c = 1.

  

  

Intercepts: f (x) and g(x) have no x-intercepts, h(x) has one at −1 because c = 1. g(x) and h(x) have one y-intercept at 1 and f(x) has one y-intercept at 2. The graphs are all shifted up 1 unit from the graph of the parent functions because c = 1. Increasing/Decreasing: f (x) and h(x) are increasing on the entire domain. g(x) is increasing to the right of the vertex and decreasing to the left.

Positive/Negative: The function values for f (x) and g (x) are all positive. The function values of h(x) are negative for x < −1 and positive for x > −1.

Maxima/Minima: f (x) and h(x) have no maxima or minima. g(x) has a minimum at (0, 1).

Symmetry: f (x) and g(x) have no symmetry. g(x) is symmetric about the y-axis.

End behavior: For f (x) and h(x), as x increases, y increases and as x decreases, y decreases. For g(x), as x increases, y increases and as x decreases, y increases. The exponential function f (x) eventually exceeds the others.

Conclusion:

The exponential function f (x) eventually exceeds the others.