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Difficulty: MediumQuadratic Functions and Graphs

In the xyxy-plane, the graph of the quadratic function f(x)=−x2+bx+cf(x) = -x^2 + bx + c, where bb and cc are constants, has its vertex at (4,25)(4, 25). If the positive xx-intercept of the graph of ff is (d,0)(d, 0), what is the value of dd?

Answer: 9

Answer

The value of dd is 99.
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. Since the vertex is (4,25)(4, 25) and the coefficient of x2x^2 is −1-1, the function is f(x)=−(x−4)2+25f(x) = -(x-4)^2 + 25. Setting f(x)=0f(x) = 0 to find the xx-intercepts yields −(x−4)2+25=0-(x-4)^2 + 25 = 0, which simplifies to (x−4)2=25(x-4)^2 = 25. Taking the square root of both sides gives x−4=5x - 4 = 5 or x−4=−5x - 4 = -5. Solving these equations gives x=9x = 9 or x=−1x = -1. The positive xx-intercept is (9,0)(9, 0), so the value of dd is 99.

Step-by-Step Solution

1
Write the function in vertex form.
f(x)=−(x−4)2+25f(x) = -(x-4)^2 + 25
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, the vertex is (4,25)(4, 25) and the coefficient of x2x^2 is −1-1, so a=−1a = -1, h=4h = 4, and k=25k = 25.
2
Set f(x)=0f(x) = 0 to find the xx-intercepts.
−(x−4)2+25=0-(x-4)^2 + 25 = 0
The xx-intercepts of a graph are the points where the function value is equal to 0.
3
Solve the equation for xx.
x=9x = 9 or x=−1x = -1
Rearranging the equation gives (x−4)2=25(x-4)^2 = 25. Taking the square root of both sides gives x−4=5x-4 = 5 or x−4=−5x-4 = -5, which solves to x=9x = 9 or x=−1x = -1.
4
Identify the positive xx-intercept coordinate dd.
d=9d = 9
The question asks for the positive xx-intercept (d,0)(d, 0), which corresponds to x=9x = 9.

Key Concept

Vertex form of a quadratic function and finding xx-intercepts
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Topics

Quadratic Functions and Graphs

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