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Difficulty: MediumLinear Functions and Graphs

The table below shows some values of xx and the corresponding values of the linear function ff.

xxf(x)f(x)
−2-21313
2255
66−3-3

If the graph of a second linear function, gg, has a slope that is twice the slope of the graph of ff and passes through the point (1,4)(1, 4), what is the value of g(5)g(5)?

  1. A
    0
  2. B
    -20
  3. -12Answer
  4. D
    -16

Answer

-12
The correct answer is −12-12. First, calculate the slope of the linear function ff using two coordinate pairs from the table, such as (−2,13)(-2, 13) and (2,5)(2, 5). The slope is 5−132−(−2)=−2\frac{5 - 13}{2 - (-2)} = -2. Since the slope of function gg is twice the slope of ff, the slope of gg is 2×(−2)=−42 \times (-2) = -4. Using the point-slope equation with the point (1,4)(1, 4), we have g(x)−4=−4(x−1)g(x) - 4 = -4(x - 1), which simplifies to g(x)=−4x+8g(x) = -4x + 8. Finally, evaluating the function at x=5x = 5 gives g(5)=−4(5)+8=−12g(5) = -4(5) + 8 = -12.

Step-by-Step Solution

1
Calculate the slope of the linear function ff using two points from the table, (−2,13)(-2, 13) and (2,5)(2, 5).
The slope of ff is mf=5−132−(−2)=−84=−2m_f = \frac{5 - 13}{2 - (-2)} = \frac{-8}{4} = -2.
Finding the slope of the reference function is necessary to determine the slope of the second function.
2
Determine the slope of the function gg.
The slope of gg is mg=2×(−2)=−4m_g = 2 \times (-2) = -4.
The problem states that the slope of gg is twice the slope of ff.
3
Find the equation of g(x)g(x) using the point-slope form with the point (1,4)(1, 4).
g(x)−4=−4(x−1)  ⟹  g(x)=−4x+8g(x) - 4 = -4(x - 1) \implies g(x) = -4x + 8.
Using a point on the line and its slope allows us to construct the full function definition.
4
Evaluate g(5)g(5) by substituting x=5x = 5 into the equation for g(x)g(x).
g(5)=−4(5)+8=−12g(5) = -4(5) + 8 = -12.
This yields the final value requested by the question.

Key Concept

Linear Functions and Graphs
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Topics

Linear Functions and Graphs

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