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Difficulty: MediumQuadratic Functions and Graphs

The quadratic function ff is defined by f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. In the xyxy-plane, the graph of ff is a parabola with vertex (3,−5)(3, -5) that passes through the point (0,4)(0, 4). What is the value of a+b+ca + b + c?

  1. A
    11
  2. B
    -9
  3. -1Answer
  4. D
    -7

Answer

-1
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. Substituting the given vertex (3,−5)(3, -5) gives f(x)=a(x−3)2−5f(x) = a(x - 3)^2 - 5. Since the graph passes through the point (0,4)(0, 4), substituting x=0x = 0 and f(x)=4f(x) = 4 yields the equation 4=a(0−3)2−54 = a(0 - 3)^2 - 5, which simplifies to 9a=99a = 9, or a=1a = 1. The function is therefore defined by f(x)=(x−3)2−5f(x) = (x - 3)^2 - 5. The expression a+b+ca + b + c represents the sum of the coefficients of the quadratic function in standard form f(x)=ax2+bx+cf(x) = ax^2 + bx + c. Evaluating the function at x=1x = 1 gives f(1)=a(1)2+b(1)+c=a+b+cf(1) = a(1)^2 + b(1) + c = a + b + c. Substituting x=1x = 1 into our vertex form equation yields f(1)=(1−3)2−5=(−2)2−5=4−5=−1f(1) = (1 - 3)^2 - 5 = (-2)^2 - 5 = 4 - 5 = -1. Therefore, the value of a+b+ca + b + c is −1-1.

Step-by-Step Solution

1
Write the quadratic function in vertex form using the given vertex (3,−5)(3, -5).
f(x)=a(x−3)2−5f(x) = a(x - 3)^2 - 5, where aa is a constant.
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.
2
Substitute the coordinates of the given point (0,4)(0, 4) into the vertex form equation to solve for aa.
4=a(0−3)2−54 = a(0 - 3)^2 - 5, which simplifies to 4=9a−54 = 9a - 5, and solving for aa gives a=1a = 1.
Since the graph passes through (0,4)(0, 4), substituting these coordinates into the function's equation must yield a true statement.
3
Find the value of a+b+ca + b + c by evaluating f(1)f(1).
f(1)=1(1−3)2−5=1(−2)2−5=4−5=−1f(1) = 1(1 - 3)^2 - 5 = 1(-2)^2 - 5 = 4 - 5 = -1.
For any quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c, evaluating the function at x=1x = 1 gives f(1)=a(1)2+b(1)+c=a+b+cf(1) = a(1)^2 + b(1) + c = a + b + c.

Key Concept

Quadratic Functions and Graphs
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Topics

Quadratic Functions and Graphs

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