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Difficulty: HardFunction Notation and Transformations

Several values of xx and the corresponding values of f(x)f(x) for the quadratic function ff are shown in the table below.

xxf(x)f(x)
−4-41818
−2-266
0022
2266
441818

The function gg is defined by g(x)=−2(f(x−3)+4)g(x) = -2(f(x - 3) + 4). If the vertex of the graph of y=f(x)y = f(x) corresponds to the point (p,q)(p, q) on the graph of y=g(x)y = g(x), what is the value of p+qp + q?

  1. A
    −15-15
  2. −9-9Answer
  3. C
    −5-5
  4. D
    33

Answer

−9-9
The vertex of the quadratic function f(x)f(x) is identified from the symmetric table values as (0,2)(0, 2). Under the transformation g(x)=−2(f(x−3)+4)g(x) = -2(f(x - 3) + 4), the input x−3x - 3 shifts the vertex horizontally to the right by 3 units, so the new x-coordinate is p=3p = 3. Evaluating g(3)g(3) gives q=−2(f(0)+4)=−2(2+4)=−2(6)=−12q = -2(f(0) + 4) = -2(2 + 4) = -2(6) = -12. The sum of these coordinates is p+q=3+(−12)=−9p + q = 3 + (-12) = -9.

Step-by-Step Solution

1
Identify the vertex of the function f(x)f(x) from the table.
Vertex of f(x)f(x) is (0,2)(0, 2).
Since ff is a quadratic function and the table shows symmetry about x=0x = 0 (with f(−2)=f(2)=6f(-2) = f(2) = 6 and f(−4)=f(4)=18f(-4) = f(4) = 18), the vertex must be at the point where x=0x = 0, which gives f(0)=2f(0) = 2.
2
Determine the x-coordinate pp of the corresponding point on the graph of g(x)g(x).
p=3p = 3.
The function gg is defined as g(x)=−2(f(x−3)+4)g(x) = -2(f(x - 3) + 4). The expression f(x−3)f(x - 3) indicates a horizontal translation of the graph of ff to the right by 3 units. Therefore, the x-coordinate of the vertex shifts from 00 to 0+3=30 + 3 = 3.
3
Determine the y-coordinate qq of the corresponding point on the graph of g(x)g(x) by evaluating g(3)g(3).
q=−12q = -12.
Substitute x=3x = 3 into the definition of g(x)g(x): g(3)=−2(f(3−3)+4)=−2(f(0)+4)g(3) = -2(f(3 - 3) + 4) = -2(f(0) + 4). Since f(0)=2f(0) = 2, this simplifies to −2(2+4)=−2(6)=−12-2(2 + 4) = -2(6) = -12.
4
Calculate the value of p+qp + q.
p+q=−9p + q = -9.
Adding the coordinates p=3p = 3 and q=−12q = -12 yields 3+(−12)=−93 + (-12) = -9.

Key Concept

Function Notation and Transformations
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Topics

Function Notation and Transformations

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