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Zorluk: Çok zorLinear Functions and Graphs

In the xyxy-plane, the graph of a linear function ff passes through the points (0,1)(0, 1) and (2,k)(2, k). The graph of another linear function gg is perpendicular to the graph of ff and passes through the points (−4,8)(-4, 8) and (2,k)(2, k). Which of the following could be the value of kk?

  1. A
    2
  2. B
    8
  3. 5Cevap
  4. D
    9

Cevap

5
The slope of the line representing function ff is calculated as mf=k−12m_f = \frac{k - 1}{2}, and the slope of the line representing function gg is mg=k−86m_g = \frac{k - 8}{6}. Since the lines are perpendicular, the product of their slopes must be −1-1. This gives the equation k−12⋅k−86=−1\frac{k - 1}{2} \cdot \frac{k - 8}{6} = -1, which simplifies to (k−1)(k−8)=−12(k - 1)(k - 8) = -12. Expanding and setting the quadratic equation to zero gives k2−9k+20=0k^2 - 9k + 20 = 0. Factoring this expression yields (k−4)(k−5)=0(k - 4)(k - 5) = 0. Thus, kk can be either 44 or 55. Since 5 is the only option listed, the option containing 5 is the correct answer.

Adım Adım Çözüm

1
Express the slope of the linear function ff in terms of kk.
mf=k−12−0=k−12m_f = \frac{k - 1}{2 - 0} = \frac{k - 1}{2}
The slope formula is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Express the slope of the linear function gg in terms of kk.
mg=k−82−(−4)=k−86m_g = \frac{k - 8}{2 - (-4)} = \frac{k - 8}{6}
The slope formula is applied to the points (−4,8)(-4, 8) and (2,k)(2, k).
3
Apply the perpendicular lines condition to set up an equation for kk.
mf⋅mg=−1  ⟹  (k−12)(k−86)=−1m_f \cdot m_g = -1 \implies \left(\frac{k - 1}{2}\right)\left(\frac{k - 8}{6}\right) = -1
Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is −1-1.
4
Solve the quadratic equation for kk.
(k−1)(k−8)=−12  ⟹  k2−9k+8=−12  ⟹  k2−9k+20=0  ⟹  (k−4)(k−5)=0(k - 1)(k - 8) = -12 \implies k^2 - 9k + 8 = -12 \implies k^2 - 9k + 20 = 0 \implies (k - 4)(k - 5) = 0
Multiply both sides by 12, expand the product, move all terms to one side, and factor the quadratic expression.
5
Identify the possible values of kk and match with the options.
k=4k = 4 or k=5k = 5. Since 5 is the only value present in the options, the correct value is 5.
Both values satisfy the condition, but only one is listed among the multiple-choice options.

Anahtar Kavram

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (their product is −1-1).
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Linear Functions and Graphs

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