Comparing Exponential, Linear, and Quadratic Growth — Quiz

Question 14 • Algebra I

For which pair of functions is the exponential consistently growing at a faster rate than the quadratic over the interval 0 is less than or equal to x comma x is less than or equal to 5 ?

Answer

Image with description On a coordinate plane, the x axis lists negative 1 to 5 by 1, and the y axis lists negative 2 to 8 by 1. The graph has two smooth curves. The first smooth curve originates from the negative direction of the x axis, intersects the y axis at (0 comma 1), goes through (1 comma 4), and moves up. The second smooth curve is a parabola that opens up and has a vertex at the origin. It goes through (2, 2) and moves up.

Image with description On a coordinate plane, the x axis lists negative 1 to 5 by 1 unit. The y axis lists negative 2 to 8 by 1 unit. The parabola opens up, has vertex at the origin, and passes through (negative 1 comma 4) and (1 comma 4). The curve goes through (0 comma 1), (1 comma 2), (2 comma 4), and (3 comma 8).

Image with description On a coordinate plane, the x axis lists negative 1 to 5 by 1 unit. The y axis lists negative 2 to 8 by 1 unit. The parabola opens up, has vertex at the origin, and passes through (negative 1 comma 2) and (1 comma 2). The curve goes through (0 comma 1), (1 comma 2), (2 comma 4), and (3 comma 8).

Image with description On a coordinate plane, the x axis lists negative 1 to 5 by 1, and the y axis lists negative 2 to 8 by 1. The graph has two smooth curves. The first smooth curve originates below the negative direction of the x axis, goes through (0 comma 0), and moves further up. The second smooth curve is a parabola that opens up and has a vertex at the origin. It goes through (1 comma 3) and moves up. Both curves touch each other at the origin.

Explanation

In choice C, the exponential passes through (0, 1) and (1, 4), so it grows by a factor of 4 for each 1-unit increase in x. The parabola has its vertex at the origin and passes through (2, 2), so its equation is y = x²/2 and its growth rate is x. The exponential’s rate is (ln 4)·4ˣ, which is greater than x throughout 0 ≤ x ≤ 5, so the exponential consistently grows faster.

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