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Difficulty: HardQuadratic Functions and Graphs

In the xyxy-plane, the graph of the quadratic function f(x)=−2(x−d)2+8f(x) = -2(x - d)^2 + 8, where dd is a positive constant, intersects the xx-axis at the point (2,0)(2, 0). The function gg is defined by g(x)=f(x+3)−5g(x) = f(x + 3) - 5. What is the yy-coordinate of the yy-intercept of the graph of gg in the xyxy-plane?

  1. A
    -15
  2. B
    -95
  3. 1Answer
  4. D
    11

Answer

The yy-coordinate of the yy-intercept of the graph of gg is 11.
To find the correct answer, we first substitute the point (2,0)(2,0) into f(x)=−2(x−d)2+8f(x) = -2(x - d)^2 + 8 to obtain 0=−2(2−d)2+80 = -2(2 - d)^2 + 8. Simplifying this equation gives (2−d)2=4(2 - d)^2 = 4, which has solutions d=0d = 0 and d=4d = 4. Since the problem states that dd is a positive constant, we have d=4d = 4, which means f(x)=−2(x−4)2+8f(x) = -2(x - 4)^2 + 8. We then find the equation for the transformed function g(x)g(x) by replacing xx with x+3x + 3 and subtracting 55, which yields g(x)=f(x+3)−5=−2(x+3−4)2+8−5=−2(x−1)2+3g(x) = f(x + 3) - 5 = -2(x + 3 - 4)^2 + 8 - 5 = -2(x - 1)^2 + 3. To find the yy-coordinate of the yy-intercept, we evaluate g(x)g(x) at x=0x = 0, giving g(0)=−2(0−1)2+3=−2(1)+3=1g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1.

Step-by-Step Solution

1
Use the given xx-intercept (2,0)(2,0) of f(x)f(x) to solve for the constant dd.
d=4d = 4
Substitute x=2x = 2 and f(x)=0f(x) = 0 into the equation f(x)=−2(x−d)2+8f(x) = -2(x - d)^2 + 8 to get 0=−2(2−d)2+80 = -2(2 - d)^2 + 8. Dividing by −2-2 and isolating the squared term gives (2−d)2=4(2 - d)^2 = 4. Taking the square root of both sides gives 2−d=22 - d = 2 or 2−d=−22 - d = -2, which yields d=0d = 0 or d=4d = 4. Since dd must be a positive constant, we must choose d=4d = 4.
2
Determine the equation of the transformed function g(x)g(x).
g(x)=−2(x−1)2+3g(x) = -2(x - 1)^2 + 3
Substitute d=4d = 4 into f(x)f(x) to get f(x)=−2(x−4)2+8f(x) = -2(x - 4)^2 + 8. Since g(x)=f(x+3)−5g(x) = f(x + 3) - 5, substitute x+3x + 3 for xx in f(x)f(x) and subtract 55: g(x)=−2((x+3)−4)2+8−5g(x) = -2((x + 3) - 4)^2 + 8 - 5, which simplifies to g(x)=−2(x−1)2+3g(x) = -2(x - 1)^2 + 3.
3
Find the yy-coordinate of the yy-intercept of the graph of gg.
11
The yy-intercept of a graph is the point where x=0x = 0. Substitute x=0x = 0 into the equation for g(x)g(x) to find g(0)=−2(0−1)2+3=−2(1)+3=1g(0) = -2(0 - 1)^2 + 3 = -2(1) + 3 = 1.

Key Concept

Applying horizontal and vertical translations to quadratic equations and identifying graph features.
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Topics

Quadratic Functions and Graphs

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