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Difficulty: Very hardFunction Notation and Transformations

The quadratic function ff is defined by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where aa, hh, and kk are constants. In the xyxy-plane, the graph of y=f(x)y = f(x) has a vertex at (3,−4)(3, -4) and passes through the point (5,8)(5, 8). If the function gg is defined by g(x)=−2f(x−1)+5g(x) = -2f(x - 1) + 5, what is the value of g(2)g(2)?

  1. A
    5
  2. B
    7
  3. -11Answer
  4. D
    13

Answer

-11
To find the value of g(2)g(2), we first determine the equation of the quadratic function f(x)f(x). Since the vertex is (3,−4)(3, -4), the vertex form is f(x)=a(x−3)2−4f(x) = a(x - 3)^2 - 4. Substituting the point (5,8)(5, 8) yields 8=a(5−3)2−48 = a(5 - 3)^2 - 4, which simplifies to 12=4a12 = 4a, so a=3a = 3. Therefore, f(x)=3(x−3)2−4f(x) = 3(x - 3)^2 - 4. We then substitute x=2x = 2 into the definition of g(x)g(x), obtaining g(2)=−2f(2−1)+5=−2f(1)+5g(2) = -2f(2 - 1) + 5 = -2f(1) + 5. Evaluating f(1)f(1) gives f(1)=3(1−3)2−4=8f(1) = 3(1 - 3)^2 - 4 = 8. Substituting this back into the expression for g(2)g(2) gives −2(8)+5=−11-2(8) + 5 = -11. Thus, the option with value -11 is correct.

Step-by-Step Solution

1
Write the quadratic function f(x)f(x) in vertex form using the given vertex (3,−4)(3, -4).
f(x)=a(x−3)2−4f(x) = a(x - 3)^2 - 4
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 point (5,8)(5, 8) into the vertex form to solve for the constant aa.
8=a(5−3)2−4  ⟹  8=4a−4  ⟹  12=4a  ⟹  a=38 = a(5 - 3)^2 - 4 \implies 8 = 4a - 4 \implies 12 = 4a \implies a = 3. Thus, f(x)=3(x−3)2−4f(x) = 3(x - 3)^2 - 4.
Since the graph of ff passes through (5,8)(5, 8), these coordinates must satisfy the function's equation.
3
Substitute x=2x = 2 into the definition of g(x)g(x) to express g(2)g(2) in terms of ff.
g(2)=−2f(2−1)+5=−2f(1)+5g(2) = -2f(2 - 1) + 5 = -2f(1) + 5
We need to evaluate the inner function transformation f(x−1)f(x - 1) at x=2x = 2.
4
Evaluate f(1)f(1) using the formula determined in Step 2.
f(1)=3(1−3)2−4=3(−2)2−4=3(4)−4=8f(1) = 3(1 - 3)^2 - 4 = 3(-2)^2 - 4 = 3(4) - 4 = 8
To find g(2)g(2), we must first compute the value of f(1)f(1).
5
Substitute f(1)=8f(1) = 8 back into the expression for g(2)g(2) and simplify.
g(2)=−2(8)+5=−16+5=−11g(2) = -2(8) + 5 = -16 + 5 = -11
This completes the evaluation of the multi-step transformation.

Key Concept

Function Notation and Transformations
Estimated Time:2m 0s
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Topics

Function Notation and Transformations

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