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Difficulty: HardFunction Notation and Transformations

A quadratic function ff has its vertex at (4,−3)(4, -3) in the coordinate plane. The point (2,5)(2, 5) is on the graph of y=f(x)y = f(x). The function gg is defined by g(x)=−f(x−3)+8g(x) = -f(x - 3) + 8. What is the value of g(8)g(8)?

Answer: 9

Answer

The correct answer is 9.
The correct answer is 9. To find this, we first establish the vertex form of the quadratic function f(x)=a(x−4)2−3f(x) = a(x - 4)^2 - 3. Substituting the point (2,5)(2, 5) gives 5=a(2−4)2−35 = a(2 - 4)^2 - 3, which simplifies to 8=4a8 = 4a, so a=2a = 2. Thus, f(x)=2(x−4)2−3f(x) = 2(x - 4)^2 - 3. To find the value of g(8)g(8), we substitute 88 into the definition of g(x)g(x), yielding g(8)=−f(8−3)+8=−f(5)+8g(8) = -f(8 - 3) + 8 = -f(5) + 8. Evaluating f(5)f(5) gives 2(5−4)2−3=−12(5 - 4)^2 - 3 = -1. Finally, substituting −1-1 back into the expression for g(8)g(8) gives −(−1)+8=9-(-1) + 8 = 9.

Step-by-Step Solution

1
Write the quadratic function f(x)f(x) in vertex form using its vertex (4,−3)(4, -3).
f(x)=a(x−4)2−3f(x) = a(x - 4)^2 - 3
The vertex form of a quadratic function with vertex (h,k)(h, k) is f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.
2
Substitute the coordinates of the point (2,5)(2, 5) into the vertex form to find the value of the constant aa.
a=2a = 2, so f(x)=2(x−4)2−3f(x) = 2(x - 4)^2 - 3
Since the point (2,5)(2, 5) lies on the graph of ff, substituting x=2x = 2 and f(2)=5f(2) = 5 allows us to solve for the vertical stretch factor aa.
3
Use the definition of g(x)g(x) to express g(8)g(8) in terms of ff.
g(8)=−f(5)+8g(8) = -f(5) + 8
Substituting x=8x = 8 into the equation g(x)=−f(x−3)+8g(x) = -f(x - 3) + 8 yields g(8)=−f(8−3)+8=−f(5)+8g(8) = -f(8 - 3) + 8 = -f(5) + 8.
4
Evaluate f(5)f(5) using the equation for f(x)f(x) found in Step 2.
f(5)=−1f(5) = -1
Substituting x=5x = 5 into f(x)=2(x−4)2−3f(x) = 2(x - 4)^2 - 3 gives 2(5−4)2−3=2(1)−3=−12(5 - 4)^2 - 3 = 2(1) - 3 = -1.
5
Substitute the value of f(5)f(5) into the expression for g(8)g(8) to find the final result.
g(8)=9g(8) = 9
Substituting f(5)=−1f(5) = -1 into g(8)=−f(5)+8g(8) = -f(5) + 8 yields −(−1)+8=1+8=9-(-1) + 8 = 1 + 8 = 9.

Key Concept

Finding the equation of a quadratic function from its vertex and a point, and evaluating transformed functions using function notation.
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Topics

Function Notation and Transformations

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