Math Problem Statement

Use the graph of y equals f left parenthesis x right parenthesis to graph the function g left parenthesis x right parenthesis equals 2 f left parenthesis x plus 1 right parenthesis plus 2. y = f(x) -8 -6 -4 -2 2 4 6 8 -8 -6 -4 -2 2 4 6 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. The graph consists of five plotted points connected by line segments from left to right. The coordinates of the plotted points are at (negative 4, 0); (negative 2, 1); (0, 0); (2, negative 1); (4, 0). . . . Question content area right Part 1 Choose the correct graph of g below. A. -8 8 -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1.The graph consists of five plotted points connected by line segments from left to right. The coordinates of the plotted points are at (negative 5, 2); (negative 3, 4); (negative 1, 2); (1, 0); (3, 2). B. -8 8 -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. The graph consists of five plotted points connected by line segments from left to right. The coordinates of the plotted points are at (negative 3, 2); (negative 1, 4); (1, 2); (3, 0); (5, 2). C. -8 8 -8 8 x y

A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. The graph consists of five plotted points connected by line segments from left to right. The coordinates of the plotted points are at (negative 5, negative 2); (negative 3, 0); (negative 1, negative 2); (1, negative 4); (3, negative 2). D. -8 8 -8 8 x y

Solution

To graph the function g(x)=2f(x+1)+2g(x) = 2f(x + 1) + 2, we need to interpret the transformations involved and apply them to the graph of f(x)f(x).

Breakdown of the transformation:

  1. Inside the function: f(x+1)f(x + 1):

    • This represents a horizontal shift of the graph of f(x)f(x) 1 unit to the left.
    • For every point (x,y)(x, y) on the graph of f(x)f(x), the corresponding point on f(x+1)f(x + 1) will be (x−1,y)(x - 1, y).

  2. Outside the function: 2f(x+1)2f(x + 1):

    • This represents a vertical stretch by a factor of 2.
    • For every point (x,y)(x, y) on the graph of f(x+1)f(x + 1), the corresponding point on 2f(x+1)2f(x + 1) will be (x,2y)(x, 2y).

  3. Adding 2: 2f(x+1)+22f(x + 1) + 2:

    • This represents a vertical shift of 2 units up.
    • For every point (x,y)(x, y) on the graph of 2f(x+1)2f(x + 1), the corresponding point on g(x)=2f(x+1)+2g(x) = 2f(x + 1) + 2 will be (x,y+2)(x, y + 2).

Step-by-step transformations:

  • Start with the original points of f(x)f(x):
    (−4,0),(−2,1),(0,0),(2,−1),(4,0)(-4, 0), (-2, 1), (0, 0), (2, -1), (4, 0).

  • Apply the horizontal shift of 1 unit left:

    • f(x+1)f(x+1):
      (−5,0),(−3,1),(−1,0),(1,−1),(3,0)(-5, 0), (-3, 1), (-1, 0), (1, -1), (3, 0).

  • Apply the vertical stretch by 2:

    • 2f(x+1)2f(x+1):
      (−5,0),(−3,2),(−1,0),(1,−2),(3,0)(-5, 0), (-3, 2), (-1, 0), (1, -2), (3, 0).

  • Apply the vertical shift of 2 units up:

    • g(x)=2f(x+1)+2g(x) = 2f(x + 1) + 2:
      (−5,2),(−3,4),(−1,2),(1,0),(3,2)(-5, 2), (-3, 4), (-1, 2), (1, 0), (3, 2).

Comparison to the options:

  • The graph with the points (−5,2),(−3,4),(−1,2),(1,0),(3,2)(-5, 2), (-3, 4), (-1, 2), (1, 0), (3, 2) corresponds to Graph A.

Thus, the correct graph for g(x)=2f(x+1)+2g(x) = 2f(x + 1) + 2 is Graph A.

If you'd like more detailed steps or help with another problem, feel free to ask!

Related questions:

  1. What would the graph of g(x)=f(x+1)−2g(x) = f(x + 1) - 2 look like?
  2. How does the horizontal shift affect the domain of the function?
  3. What is the effect of a vertical stretch by a factor of 3 on the graph?
  4. How would a reflection of the function f(x)f(x) over the x-axis look?
  5. How does a combination of multiple transformations affect the graph?

Tip: When dealing with transformations, always focus on the order of operations. Horizontal transformations (like shifts) occur first, followed by vertical transformations (like stretches and shifts).

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Graphing Functions
Algebra

Formulas

g(x) = a * f(bx + c) + d

Theorems

Transformation Rules for Functions

Suitable Grade Level

Grades 9-12

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