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An exponential function and a quadratic function are graphed below. Which of the following is true of the growth rate of the functions over the interval


a.The exponential grows at half the rate of the quadratic.
b.The exponential grows at the same rate as the quadratic.
c.The exponential grows at twice the rate of the quadratic.
d.The exponential grows at four times the rate of the quadratic.

An exponential function and a quadratic function are graphed below Which of the following is true of the growth rate of the functions over the interval aThe exp class=
An exponential function and a quadratic function are graphed below Which of the following is true of the growth rate of the functions over the interval aThe exp class=

Respuesta :

nobillionaireNobley
Given an exponential function, say f(x), such that f(0) = 1 and f(1) = 2 and a quadratic finction, say g(x), such that g(0) = 0 and g(1) = 1.

The rate of change of a function f(x) over an interval
[tex]a \leq x \leq b[/tex]
is given by
[tex] \frac{f(b)-f(a)}{b-a} [/tex]

Thus, the rate of change (growth rate) of the exponential function, f(x) over the interval
[tex]0 \leq x \leq 1[/tex]
is given by
[tex] \frac{f(1)-f(0)}{1-0} = \frac{2-1}{1} =1[/tex]

Similarly, the rate of change (growth rate) of the quadratic function, g(x) over the interval
[tex]0 \leq x \leq 1[/tex]
is given by
[tex] \frac{g(1)-g(0)}{1-0} = \frac{1-0}{1} =1[/tex]

Therefore, the exponential grows at the same rate as the quadratic in the interval [tex]0 \leq x \leq 1[/tex].

The growth rate of considered function over given interval is: Option b.The exponential grows at the same rate as the quadratic.

How to measure the rate of change of something as some other value changes?

Suppose that we have to measure the rate of change of y as x changes, then we have:

[tex]Rate = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]

where we have

[tex]\rm when \: x=x_1, y = y_1\\when\: x = x_2, y= y_2[/tex]

Remember that, we divide by the change in independent variable so that we get some idea of how much the dependent quantity changes as we change the independent quantity by 1 unit.

(5 change per 3 unit can be rewritten as 5/3 change per 1 unit)

For the given case, the interval in consideration is [tex]0 \leq x \leq 1[/tex]

The input change is

[tex]x_1 = 0 \\to \\x_2 = 1[/tex]

For it, the output of functions is changing as:

  • For exponential:

[tex]y_1 = 1[/tex] to [tex]y_2 = 2[/tex]

Thus, rate =  [tex]\dfrac{2-1}{1-0} = 1[/tex]

  • For quadratic:

[tex]y_1 = 0[/tex]  to   [tex]y_2 = 1[/tex]

Thus, rate = [tex]\dfrac{1-0}{1-0} = 1[/tex]

Thus, The growth rate of considered function over given interval is: Option b.The exponential grows at the same rate as the quadratic.

Learn more about rate of growth of function here:

https://brainly.com/question/15125607

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