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Difficulty: HardQuadratic Functions and Graphs

A parabola in the xyxy-plane has a vertex at (3,−4)(3, -4) and passes through the point (1,0)(1, 0). A second parabola is created by reflecting the original parabola across the xx-axis, then translating the resulting graph 44 units to the left and 22 units down. If this second parabola represents the graph of the function gg, which of the following equations defines g(x)g(x)?

  1. A
    g(x)=−(x−7)2+2g(x) = -(x - 7)^2 + 2
  2. g(x)=−(x+1)2+2g(x) = -(x + 1)^2 + 2Answer
  3. C
    g(x)=−(x+1)2−6g(x) = -(x + 1)^2 - 6
  4. D
    g(x)=(x+1)2−6g(x) = (x + 1)^2 - 6

Answer

The equation g(x)=−(x+1)2+2g(x) = -(x + 1)^2 + 2
The correct equation is g(x)=−(x+1)2+2g(x) = -(x + 1)^2 + 2. Since the vertex of the original parabola f(x)f(x) is (3,−4)(3, -4) and it passes through (1,0)(1, 0), its vertex form equation is f(x)=(x−3)2−4f(x) = (x - 3)^2 - 4. Reflecting f(x)f(x) across the xx-axis negates the entire function, resulting in −(x−3)2+4-(x - 3)^2 + 4. Shifting this function 44 units to the left replaces xx with x+4x + 4, yielding −(x+1)2+4-(x + 1)^2 + 4. Finally, shifting it 22 units down subtracts 22 from the entire expression, giving g(x)=−(x+1)2+2g(x) = -(x + 1)^2 + 2.

Step-by-Step Solution

1
Determine the equation of the original parabola f(x)f(x) using its vertex form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k.
f(x)=(x−3)2−4f(x) = (x - 3)^2 - 4
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 (1,0)(1, 0) gives 0=a(1−3)2−4  ⟹  4a=4  ⟹  a=10 = a(1 - 3)^2 - 4 \implies 4a = 4 \implies a = 1.
2
Reflect the graph of f(x)f(x) across the xx-axis.
−f(x)=−(x−3)2+4-f(x) = -(x - 3)^2 + 4
A reflection across the xx-axis negates the entire function, so y=−f(x)y = -f(x).
3
Translate the reflected graph 44 units to the left.
−(x+1)2+4-(x + 1)^2 + 4
Translating a function h(x)h(x) to the left by cc units is represented by h(x+c)h(x + c). Replacing xx with x+4x + 4 in −(x−3)2+4-(x - 3)^2 + 4 gives −((x+4)−3)2+4=−(x+1)2+4-((x + 4) - 3)^2 + 4 = -(x + 1)^2 + 4.
4
Translate the graph 22 units down to find the final function g(x)g(x).
g(x)=−(x+1)2+2g(x) = -(x + 1)^2 + 2
Translating a function down by dd units is represented by subtracting dd from the function. Subtracting 22 from −(x+1)2+4-(x + 1)^2 + 4 gives −(x+1)2+2-(x + 1)^2 + 2.

Key Concept

Applying transformations (reflections and horizontal/vertical translations) to quadratic functions in vertex form.
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Topics

Quadratic Functions and Graphs

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