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Difficulty: Very hardQuadratic Equations and Factoring

A parabola defined by the quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where a,b,a, b, and cc are real constants with a>0a > 0, has its vertex at a minimum value of −16-16. The distance between the two xx-intercepts of the parabola is 88. If f(1)=−7f(1) = -7 and the xx-coordinate of the vertex is positive, what is the value of f(−2)f(-2)?

Answer: 20

Answer

The value of f(−2)f(-2) is 20.
By converting the parabola into vertex form f(x)=a(x−h)2−16f(x) = a(x - h)^2 - 16, the xx-intercepts are found at h±4ah \pm \frac{4}{\sqrt{a}}. Equating their difference to 88 yields a=1a = 1. Substituting f(1)=−7f(1) = -7 gives (1−h)2=9(1 - h)^2 = 9, which yields h=4h = 4 under the condition h>0h > 0. Evaluating f(−2)=(−2−4)2−16f(-2) = (-2 - 4)^2 - 16 produces 2020.

Step-by-Step Solution

1
Express the quadratic function in vertex form using the minimum value
f(x)=a(x−h)2−16f(x) = a(x - h)^2 - 16, where (h,−16)(h, -16) is the vertex.
Since a>0a > 0, the parabola opens upwards and its minimum value occurs at the vertex yy-coordinate, k=−16k = -16.
2
Determine the leading coefficient aa using the distance between xx-intercepts
a=1a = 1
Setting f(x)=0f(x) = 0 yields a(x−h)2−16=0  ⟹  (x−h)2=16a  ⟹  x=h±4aa(x - h)^2 - 16 = 0 \implies (x - h)^2 = \frac{16}{a} \implies x = h \pm \frac{4}{\sqrt{a}}. The distance between roots is 8a=8\frac{8}{\sqrt{a}} = 8, which gives a=1  ⟹  a=1\sqrt{a} = 1 \implies a = 1.
3
Determine the vertex xx-coordinate hh using the point f(1)=−7f(1) = -7
h=4h = 4
Substituting a=1a = 1 and x=1x = 1 into the vertex form gives (1−h)2−16=−7  ⟹  (1−h)2=9(1 - h)^2 - 16 = -7 \implies (1 - h)^2 = 9. Taking square roots gives 1−h=3  ⟹  h=−21 - h = 3 \implies h = -2 or 1−h=−3  ⟹  h=41 - h = -3 \implies h = 4. Since h>0h > 0, we select h=4h = 4.
4
Evaluate f(−2)f(-2) using the fully specified function
f(−2)=20f(-2) = 20
With f(x)=(x−4)2−16f(x) = (x - 4)^2 - 16, substituting x=−2x = -2 yields f(−2)=(−2−4)2−16=(−6)2−16=36−16=20f(-2) = (-2 - 4)^2 - 16 = (-6)^2 - 16 = 36 - 16 = 20.

Key Concept

Quadratic Vertex Form, Root Separation, and Evaluation
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Topics

Quadratic Equations and Factoring

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