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Difficulty: MediumLinear Functions and Graphs

In the xyxy-plane, the graph of the linear function ff passes through the points (−2,5)(-2, 5) and (4,1)(4, 1). If the linear function gg is defined by g(x)=f(x)+3g(x) = f(x) + 3, what is the xx-intercept of the graph of gg?

  1. A
    66
  2. B
    112\frac{11}{2}
  3. 1010Answer
  4. D
    203\frac{20}{3}

Answer

1010
The correct answer is 1010. First, find the slope of ff using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, which gives m=1−54−(−2)=−23m = \frac{1 - 5}{4 - (-2)} = -\frac{2}{3}. Using the point (4,1)(4, 1) in the point-slope formula, the equation of ff is f(x)=−23x+113f(x) = -\frac{2}{3}x + \frac{11}{3}. Since g(x)=f(x)+3g(x) = f(x) + 3, the equation of gg is g(x)=−23x+203g(x) = -\frac{2}{3}x + \frac{20}{3}. Setting g(x)=0g(x) = 0 to find the xx-intercept gives x=10x = 10.

Step-by-Step Solution

1
Calculate the slope of the linear function ff
m=−23m = -\frac{2}{3}
The slope mm of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}. Substituting (−2,5)(-2, 5) and (4,1)(4, 1) yields m=1−54−(−2)=−46=−23m = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}.
2
Find the equation of function ff
f(x)=−23x+113f(x) = -\frac{2}{3}x + \frac{11}{3}
Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) with the point (4,1)(4, 1), we get y−1=−23(x−4)⇒y=−23x+83+1=−23x+113y - 1 = -\frac{2}{3}(x - 4) \Rightarrow y = -\frac{2}{3}x + \frac{8}{3} + 1 = -\frac{2}{3}x + \frac{11}{3}.
3
Determine the equation of function gg
g(x)=−23x+203g(x) = -\frac{2}{3}x + \frac{20}{3}
Since g(x)=f(x)+3g(x) = f(x) + 3, we add 33 to the expression for f(x)f(x): g(x)=−23x+113+3=−23x+203g(x) = -\frac{2}{3}x + \frac{11}{3} + 3 = -\frac{2}{3}x + \frac{20}{3}.
4
Find the xx-intercept of the graph of gg
x=10x = 10
The xx-intercept is the value of xx when g(x)=0g(x) = 0. Setting g(x)=0g(x) = 0 gives 0=−23x+203⇒23x=203⇒x=100 = -\frac{2}{3}x + \frac{20}{3} \Rightarrow \frac{2}{3}x = \frac{20}{3} \Rightarrow x = 10.

Key Concept

Finding the equation of a linear function from two points and applying vertical translations to find key features such as intercepts.
Estimated Time:1m 30s
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Topics

Linear Functions and Graphs

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