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Difficulty: MediumQuadratic Functions and Graphs

The graph of the quadratic function ff in the xyxy-plane has its vertex at (3,2)(3, 2). If the graph passes through the point (5,−6)(5, -6), what is the yy-intercept of the graph of ff?

  1. −16-16Answer
  2. B
    −11-11
  3. C
    −1-1
  4. D
    22

Answer

The y-intercept of the graph of ff is −16-16.
The correct answer is found by setting up the vertex form of the quadratic function, f(x)=a(x−3)2+2f(x) = a(x-3)^2 + 2, using the vertex (3,2)(3, 2). Substituting the point (5,−6)(5, -6) yields −6=a(2)2+2-6 = a(2)^2 + 2, which simplifies to −8=4a-8 = 4a, giving a=−2a = -2. The complete equation is f(x)=−2(x−3)2+2f(x) = -2(x-3)^2 + 2. Evaluating this function at x=0x = 0 yields −2(0−3)2+2=−18+2=−16-2(0-3)^2 + 2 = -18 + 2 = -16.

Step-by-Step Solution

1
Write the quadratic function in vertex form.
f(x)=a(x−3)2+2f(x) = a(x-3)^2 + 2
The vertex form of a quadratic function is f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h, k) is the vertex. Here, (h,k)=(3,2)(h, k) = (3, 2).
2
Substitute the coordinates of the given point (5,−6)(5, -6) to solve for the coefficient aa.
a=−2a = -2
Plugging x=5x=5 and f(x)=−6f(x)=-6 into the equation gives −6=a(5−3)2+2-6 = a(5-3)^2 + 2, which simplifies to −6=4a+2-6 = 4a + 2. Subtracting 22 from both sides gives −8=4a-8 = 4a, so a=−2a = -2.
3
Substitute a=−2a = -2 back into the vertex form to write the full equation of the function.
f(x)=−2(x−3)2+2f(x) = -2(x-3)^2 + 2
This represents the specific quadratic function described.
4
Evaluate the function at x=0x = 0 to find the y-intercept.
f(0)=−16f(0) = -16
The y-intercept of a graph is the point where x=0x=0. Evaluating f(0)f(0) gives −2(0−3)2+2=−2(9)+2=−16-2(0-3)^2 + 2 = -2(9) + 2 = -16.

Key Concept

Writing a quadratic function in vertex form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k and finding the y-intercept by evaluating the function at x=0x=0.
Estimated Time:1m 30s
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Topics

Quadratic Functions and Graphs

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