Math Problem Statement

se the graph of y equals f left parenthesis x right parenthesis shown to the right to graph the following function g. g left parenthesis x right parenthesis equals negative one half f left parenthesis x minus 1 right parenthesis minus 4 -10 10 -10 4 x y y equals f left parenthesis x right parenthesisleft parenthesis 0 comma 0 right parenthesisleft parenthesis negative 5 comma 0 right parenthesisleft parenthesis 0 comma negative 4 right parenthesisleft parenthesis 5 comma negative 4 right parenthesis

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 4 in increments of 1. A line segment connects the labeled points (0, 0) and (5, negative 4). A second line segment connects the labeled points (negative 5, 0) and (0, negative 4). Question content area bottom Part 1 Choose the correct graph below. A. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A line segment connects the labeled points (negative 4, 0) and (1, 4). A second line segment connects the labeled points (1, 0) and (6, 4). B. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A line segment connects the labeled points (negative 6, 0) and (negative 1, 2). A second line segment connects the labeled points (negative 1, 0) and (4, 2). C. -10 10 -10 10 x y

A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A line segment connects the labeled points (negative 4, negative 4) and (1, negative 2). A second line segment connects the labeled points (1, negative 4) and (6, negative 2). D. -10 10 -10 10 x y

Solution

We are tasked with graphing the function g(x)=−12f(x−1)−4g(x) = -\frac{1}{2} f(x - 1) - 4 based on the graph of f(x)f(x) given. Let’s break down how the transformations affect the graph step by step:

Steps to Transform f(x)f(x) into g(x)g(x):

  1. Horizontal Shift: The term f(x−1)f(x - 1) means a shift to the right by 1 unit.
  2. Vertical Stretch and Reflection: The term −12f(x−1)-\frac{1}{2} f(x - 1) means two things:
    • The negative sign reflects the graph over the x-axis.
    • The 12\frac{1}{2} scales the graph vertically by a factor of 12\frac{1}{2}, meaning the y-values will be halved.
  3. Vertical Shift: The −4-4 at the end shifts the graph down by 4 units.

Step-by-Step Explanation:

  • The graph of f(x)f(x) consists of two line segments: one from (−5,0)(-5, 0) to (0,−4)(0, -4) and another from (0,0)(0, 0) to (5,−4)(5, -4).

  • First, we shift the graph of f(x)f(x) right by 1 unit. This changes the points as follows:

    • (−5,0)(-5, 0) becomes (−4,0)(-4, 0)
    • (0,−4)(0, -4) becomes (1,−4)(1, -4)
    • (0,0)(0, 0) becomes (1,0)(1, 0)
    • (5,−4)(5, -4) becomes (6,−4)(6, -4)

  • Next, we apply the vertical reflection and scaling by −12-\frac{1}{2}, which modifies the y-values:

    • The y-value of 0 stays 0.
    • The y-value of −4-4 becomes 22 (because −12×(−4)=2-\frac{1}{2} \times (-4) = 2).

  • Finally, we shift the graph down by 4 units, changing the points again:

    • (−4,0)(-4, 0) becomes (−4,−4)(-4, -4)
    • (1,−4)(1, -4) becomes (1,−2)(1, -2)
    • (1,0)(1, 0) becomes (1,−4)(1, -4)
    • (6,−4)(6, -4) becomes (6,−2)(6, -2)

Result:

The transformed graph should have points:

  • (−4,−4)(-4, -4)
  • (1,−2)(1, -2)
  • (1,−4)(1, -4)
  • (6,−2)(6, -2)

Conclusion:

Based on the transformations, Option C correctly shows the graph of g(x)=−12f(x−1)−4g(x) = -\frac{1}{2} f(x - 1) - 4.

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Math Problem Analysis

Mathematical Concepts

Function Transformations
Graphing
Linear Functions

Formulas

g(x) = -1/2 f(x - 1) - 4

Theorems

Transformation of Functions
Vertical and Horizontal Shifts
Reflections and Scaling

Suitable Grade Level

Grades 9-11

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