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The diagram shows the graphs with equations $y = f(x)$ and $y = kf(x) + a$ - Scottish Highers Maths - Question 10 - 2022

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Question 10

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The diagram shows the graphs with equations $y = f(x)$ and $y = kf(x) + a$. (a) State the value of $a$. (b) Find the value of $k$.

Worked Solution & Example Answer:The diagram shows the graphs with equations $y = f(x)$ and $y = kf(x) + a$ - Scottish Highers Maths - Question 10 - 2022

Step 1

State the value of a.

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Answer

To find the value of aa, observe the vertical shift of the graph. The graph y=kf(x)+ay = kf(x) + a has its intercept at (0,3)(0, 3), which indicates that a=3a = 3.

Step 2

Find the value of k.

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Answer

To determine the value of kk, we need to analyze the scaling of the graph. Noticing that the point (2,−1)(2, -1) on the graph of y=f(x)y = f(x) corresponds to the point (2,5)(2, 5) on the graph of y=kf(x)+ay = kf(x) + a, we can solve:

  1. Start with the original point: f(2)=−1f(2) = -1.

  2. We know (2,5)(2, 5) belongs to y=kf(x)+ay = kf(x) + a, therefore:

    5=k(−1)+35 = k(-1) + 3

    Rearranging gives:

    k(−1)=5−3k(-1) = 5 - 3 k(−1)=2k(-1) = 2 k=−2k = -2

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