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Common Core Standard HSF-LE.A.1 Questions

Distinguish between situations that can be modeled with linear functions and with exponential functions.

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Grade 11 Linear Equations CCSS: HSF-LE.A.1, HSF-LE.A.1a
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.
Grade 11 Linear Equations CCSS: HSF-LE.A.1, HSF-LE.A.1a
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.
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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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]
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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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.
Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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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].
Grade 11 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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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]
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