# 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

- linear.
- exponential.
- logarithmic.
- none of the above.

2.

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

- Linear
- Exponential
- Both a and b
- 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] |

- [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.

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] |

- 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.

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.

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

- 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.

C.

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].

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.

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.

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

B.

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.

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?

- Equal factors are [math]x_0[/math], equal intervals are [math]c[/math].
- Equal factors are [math]alpha[/math], equal intervals are [math]x[/math].
- Equal factors are [math]alpha[/math], equal intervals are [math]c[/math].
- Equal factors are [math]c[/math], equal intervals are [math]alpha[/math].

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