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Linear and Exponential Growth (Grades 11-12)

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Linear and Exponential Growth

1. 
A function which has a constant difference per interval is
  1. linear.
  2. exponential.
  3. logarithmic.
  4. none of the above.
2. 
Which describes a function that has a constant factor per interval?
  1. Linear
  2. Exponential
  3. Both a and b
  4. None of the above
3. 
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]
  1. [math]f(x)[/math] is linear, because the difference of y-values over equal intervals is constant.
  2. [math]f(x)[/math] is linear, because the difference of x-values is constantly 2 units.
  3. [math]f(x)[/math] is exponential, because the ratio of y-values over equal intervals is constant.
  4. 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.
4. 
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]
  1. The function is linear, since the difference in most x-values is 3 units.
  2. The function is exponential, since the ratio of y-values, over equal intervals, is constant.
  3. It cannot be determined, since the difference in x-values is not constant.
  4. 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.
5. 
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].
A. 
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.
  1. [math]m Delta x + 2b[/math]
  2. [math]Delta x - 2b[/math]
  3. [math]m Delta x[/math]
  4. [math]Delta x + 2m[/math]
B. 
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?
  1. Because the resulting equation for [math]Delta f[/math] is also linear, it will increase at a constant rate.
  2. Because there is no slope in the resulting equation, the value of [math]Delta f[/math] is constant.
  3. 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.
  4. 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.
C. 
What if, for the linear function [math]f(x)[/math], [math]m =0 ?[/math]
  1. This means that the above reasoning is invalid.
  2. For how the function is defined above, [math]m[/math] cannot equal zero.
  3. It makes things easier, since the growth rate of [math]f(x)[/math] simply becomes zero (and thus constant) for all intervals.
  4. It does not change anything, since [math]Delta f[/math] is not dependent on [math]m[/math].
6. 
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.
A. 
Using the definition of [math]f(x)[/math] in the proposition, rewrite [math]f(x + \alpha) / f(x)[/math], in simplest terms.
  1. [math]2ab^{x+\alpha}[/math]
  2. [math]b^{x+2 \alpha}[/math]
  3. [math]b^alpha[/math]
  4. [math]\alpha[/math]
B. 
Which of the following reasons best explains why proposition A is true?
  1. Since [math]\alpha[/math] is a constant value, the value of [math]f(x+\alpha)/f(x)[/math] will also be a constant value.
  2. 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.
  3. Since the simplified form of [math]f(x+\alpha)/f(x)[/math] is independent of [math]x[/math], it will be a constant value.
  4. 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.
C. 
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?
  1. Equal factors are [math]x_0[/math], equal intervals are [math]c[/math].
  2. Equal factors are [math]alpha[/math], equal intervals are [math]x[/math].
  3. Equal factors are [math]alpha[/math], equal intervals are [math]c[/math].
  4. Equal factors are [math]c[/math], equal intervals are [math]alpha[/math].

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