Math Problem Statement

Answer each of the following questions about the graph for

y=4/11|x-7|-2

  1. Does the graph open up or down? Select an answer

  2. Give the vertex as an ordered pair.

  3. Is this point the maximum or minimum of the graph? Select an answer

  4. Is the graph wider, narrower, or the same width as the parent function ? Select an answer

  5. From the parent fucntion this graph has been shift horizontally ? unit(s) to the Select an answer

  6. From the parent function this graph has been shifted vertically ? unit(s) Select an answer .

Solution

Let's break down the equation step by step:

Given equation:

y=411x72y = \frac{4}{11} |x - 7| - 2

  1. Does the graph open up or down?

    • The coefficient of the absolute value function, 411\frac{4}{11}, is positive. This means the graph opens up.

    Answer: The graph opens up.

  2. Give the vertex as an ordered pair.

    • The general form of the absolute value function is y=axh+ky = a|x-h| + k, where the vertex is at (h,k)(h, k).
    • In this case, the vertex is at (7,2)(7, -2).

    Answer: The vertex is (7,2)(7, -2).

  3. Is this point the maximum or minimum of the graph?

    • Since the graph opens upwards, the vertex represents the minimum point of the graph.

    Answer: This point is the minimum.

  4. Is the graph wider, narrower, or the same width as the parent function y=xy = |x|?

    • The coefficient 411\frac{4}{11} is less than 1, which means the graph is wider than the parent function.

    Answer: The graph is wider.

  5. From the parent function y=xy = |x|, this graph has been shifted horizontally how many unit(s) to the left or right?

    • The term x7|x - 7| indicates a horizontal shift 7 units to the right.

    Answer: 7 units to the right.

  6. From the parent function y=xy = |x|, this graph has been shifted vertically how many unit(s)?

    • The 2-2 outside the absolute value indicates a vertical shift 2 units down.

    Answer: 2 units down.

Relative Questions:

  1. How would the graph change if the coefficient of the absolute value were negative?
  2. What is the effect of increasing the coefficient of x7|x-7|?
  3. How does the vertex form help identify transformations of the parent function?
  4. If the horizontal shift were 5 units to the left, what would the equation look like?
  5. How would the graph change if the vertical shift was 3 units up?

Tip:

When dealing with absolute value functions, always check the sign of the coefficient in front of the absolute value to determine the direction of the opening.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Graph Transformations

Formulas

y = a|x - h| + k (Vertex Form)
Transformation: Vertical and Horizontal Shifts
Width change based on coefficient

Theorems

Graph transformation rules for absolute value functions
Vertex determination from the equation

Suitable Grade Level

Grades 9-11