Linear and Exponential Growth (Grades 11-12)

Print Answer Key PDF Take Now Schedule Copy

Print Test (Only the test content will print)

Name:			Date:

Linear and Exponential Growth

A function which has a constant difference per interval is

linear.
exponential.
logarithmic.
none of the above.

Which describes a function that has a constant factor per interval?

Linear
Exponential
Both a and b
None of the above

Given the table below, which lists some of the values of the function [math]f(x)[/math], which of the following is true, and why?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 0 [/math]	[math] -4 [/math]
[math] 2 [/math]	[math] 2 [/math]
[math] 4 [/math]	[math] 8 [/math]
[math] 6 [/math]	[math] 14 [/math]
[math] 8 [/math]	[math] 20 [/math]

[math]f(x)[/math] is linear, because the difference of y-values over equal intervals is constant.
[math]f(x)[/math] is linear, because the difference of x-values is constantly 2 units.
[math]f(x)[/math] is exponential, because the ratio of y-values over equal intervals is constant.
It cannot be determined whether [math]f(x)[/math] is linear or exponential, because there are no intervals of only one unit in the table.

Given the table below, which lists some of the values of the function [math]f(x)[/math], which of the following is true, and why?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math]	[math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] -3 [/math]	[math] 3/8 [/math]
[math] 0 [/math]	[math] 3 [/math]
[math] 3 [/math]	[math] 24 [/math]
[math] 6 [/math]	[math] 192 [/math]
[math] 7 [/math]	[math] 384 [/math]

The function is linear, since the difference in most x-values is 3 units.
The function is exponential, since the ratio of y-values, over equal intervals, is constant.
It cannot be determined, since the difference in x-values is not constant.
The function is neither linear nor exponential. Both the difference of y-values and the ratio of y-values are not constant for all the values presented in the table.

Let [math]f(x) = mx+b[/math] be a real-valued linear function. The following questions will investigate the growth of this linear function, or the value [math]Delta f = f(x + Delta x) - f(x)[/math], over equal intervals of length [math]Delta x[/math], where [math]Delta x > 0[/math].

What is the value of [math]Delta f = f(x + Delta x) - f(x)[/math], using the definition of [math]f(x) ?[/math] Simplify your answer fully.

[math]m Delta x + 2b[/math]
[math]Delta x - 2b[/math]
[math]m Delta x[/math]
[math]Delta x + 2m[/math]

Given the answer in the previous question, which of the following gives the best reasoning as to why one can conclude that a linear function grows by equal differences over equal intervals?

Because the resulting equation for [math]Delta f[/math] is also linear, it will increase at a constant rate.
Because there is no slope in the resulting equation, the value of [math]Delta f[/math] is constant.
Since the resulting equation for [math]Delta f[/math] has no [math]b[/math] value, it is independent of the y-intercept. As such, [math]Delta f[/math] will increase by equal amounts over equal intervals.
Since [math]Delta f[/math] is dependent only on the length of the interval, as long as the interval [math]Delta x[/math] is constant, [math]Delta f[/math] will be the same.

What if, for the linear function [math]f(x)[/math], [math]m =0 ?[/math]

This means that the above reasoning is invalid.
For how the function is defined above, [math]m[/math] cannot equal zero.
It makes things easier, since the growth rate of [math]f(x)[/math] simply becomes zero (and thus constant) for all intervals.
It does not change anything, since [math]Delta f[/math] is not dependent on [math]m[/math].

Proposition A:

If [math]f[/math] is an exponential function of the form [math]f(x) = a b^x[/math], [math]a>0, \ b>0 " and " b!=1[/math], and [math]\alpha > 0[/math] is a given constant, then [math](f(x+alpha))/f(x) = c[/math] for all values of [math]x in RR[/math], where [math]c[/math] is a real-valued constant.

Using the definition of [math]f(x)[/math] in the proposition, rewrite [math]f(x + \alpha) / f(x)[/math], in simplest terms.

[math]2ab^{x+\alpha}[/math]
[math]b^{x+2 \alpha}[/math]
[math]b^alpha[/math]
[math]\alpha[/math]

Which of the following reasons best explains why proposition A is true?

Since [math]\alpha[/math] is a constant value, the value of [math]f(x+\alpha)/f(x)[/math] will also be a constant value.
Because the simplified form of [math]f(x+\alpha)/f(x)[/math] is an exponential equation, it is valid for all values of [math]x in RR[/math], and will be constant.
Since the simplified form of [math]f(x+\alpha)/f(x)[/math] is independent of [math]x[/math], it will be a constant value.
Because a constant value to the power of a constant value is also a constant, the value [math]f(x+\alpha)/f(x)[/math] will also be a constant value.

One way to restate Proposition A is as follows: "Exponential functions grow by equal factors over equal intervals." What is the value of the equal factors and length of the equal intervals as represented in Proposition A?