Examkin

Question

Difficulty: HardFunction Notation and Transformations

The graph of y=f(x)y = f(x) has a relative minimum at the point (2,−4)(2, -4) in the xyxy-plane. If the function gg is defined by g(x)=3−f(2x−6)g(x) = 3 - f(2x - 6), what are the coordinates of the corresponding relative maximum on the graph of y=g(x)y = g(x)?

  1. (4,7)(4, 7)Answer
  2. B
    (−1,7)(-1, 7)
  3. C
    (4,−1)(4, -1)
  4. D
    (6,7)(6, 7)

Answer

(4,7)(4, 7)
To find the coordinates of the relative maximum on the graph of y=g(x)y = g(x) that corresponds to the relative minimum of y=f(x)y = f(x) at (2,−4)(2, -4), we first find the new xx-coordinate. The input to the function ff must be 22, so we set the argument 2x−6=22x - 6 = 2, which solves to x=4x = 4. Next, we find the new yy-coordinate by evaluating g(4)=3−f(2)g(4) = 3 - f(2). Substituting f(2)=−4f(2) = -4, we get g(4)=3−(−4)=7g(4) = 3 - (-4) = 7. Thus, the coordinates of the corresponding relative maximum are (4,7)(4, 7).

Step-by-Step Solution

1
Set the argument of the function ff in g(x)g(x) equal to the xx-coordinate of the known point on the graph of ff.
2x−6=22x - 6 = 2
The relative minimum of ff occurs when its input is 2.
2
Solve the equation for xx to find the transformed xx-coordinate.
2x=8  ⟹  x=42x = 8 \implies x = 4
Isolating the variable xx by adding 6 to both sides and then dividing by 2.
3
Substitute x=4x = 4 into the definition of g(x)g(x) to calculate the transformed yy-coordinate.
g(4)=3−f(2(4)−6)=3−f(2)g(4) = 3 - f(2(4) - 6) = 3 - f(2)
This evaluates the vertical transformation of the function at the corresponding xx-value.
4
Substitute the value f(2)=−4f(2) = -4 and simplify.
g(4)=3−(−4)=3+4=7g(4) = 3 - (-4) = 3 + 4 = 7
Since the minimum of ff is at (2,−4)(2, -4), we know f(2)=−4f(2) = -4. The negation in front of ff reflects the minimum to a maximum.

Key Concept

Function Notation and Transformations
Estimated Time:2m 0s
Rate this question

Topics

Function Notation and Transformations

More questions from this topic