Math Problem Statement

Below is the graph of y = |x|. Translate it to make it the graph of y = |x + 5| + 2.

Solution

The problem in the image asks to translate the graph of y=xy = |x| into y=x+5+2y = |x + 5| + 2. Let's break this down step by step.

  1. Horizontal Shift:

    • The term x+5|x + 5| involves shifting the graph of y=xy = |x| 5 units to the left (opposite the sign inside the absolute value).
  2. Vertical Shift:

    • Adding +2+2 at the end means shifting the graph 2 units upward.

Translation Steps:

  1. Start with the graph of y=xy = |x|.
  2. Shift it 5 units to the left.
  3. Then shift it 2 units upward.

This will result in the graph of y=x+5+2y = |x + 5| + 2.

Would you like a detailed explanation of graph transformations or examples of similar problems?

Here are 5 related questions for further exploration:

  1. How would the graph change if the equation was y=x5+2y = |x - 5| + 2?
  2. What happens to the graph if +2+2 is replaced with 3-3?
  3. How does multiplying the absolute value, as in y=2xy = 2|x|, affect the graph?
  4. Can you explain the general formula y=axh+ky = a|x - h| + k?
  5. How would y=x+5+2y = -|x + 5| + 2 look compared to the current equation?

Tip: Always address horizontal transformations before vertical transformations for clarity in graph shifts.

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Math Problem Analysis

Mathematical Concepts

Graph Transformations
Absolute Value Functions
Horizontal and Vertical Shifts

Formulas

y = |x|
y = |x + h| + k

Theorems

Graph Translation Theorem

Suitable Grade Level

Grades 8-10