Math Problem Statement

Use transformations of the absolute value​ function,

f left parenthesis x right parenthesis equals StartAbsoluteValue x EndAbsoluteValuef(x)=x​,

to graph the function

h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 1 EndAbsoluteValue plus 3h(x)=x+1+3.

Solution

To graph the function h(x)=x+1+3h(x) = |x + 1| + 3 using transformations of the basic absolute value function f(x)=xf(x) = |x|, we need to understand how each transformation modifies the original graph. Here's how to proceed:

Step-by-Step Explanation:

  1. Start with the basic absolute value function:

    • The basic graph of f(x)=xf(x) = |x| is a V-shaped graph that has a vertex at the origin (0, 0) and opens upwards.
  2. Apply the transformation x+1x + 1 inside the absolute value**:

    • The transformation x+1|x + 1| represents a horizontal shift. Since the "+1" is inside the absolute value function, it shifts the graph 1 unit to the left. This moves the vertex from (0, 0) to (-1, 0).
  3. Apply the vertical shift of +3:

    • The "+3" outside of the absolute value function shifts the graph upwards by 3 units. This moves the vertex from (-1, 0) to (-1, 3).

Final Graph:

  • The resulting graph of h(x)=x+1+3h(x) = |x + 1| + 3 is a V-shaped graph with a vertex at the point (-1, 3).
  • The slope of the arms of the V remains the same as that of the basic absolute value function, with the left arm having a slope of -1 and the right arm having a slope of +1.

Summary of Transformations:

  1. Horizontal shift: 1 unit to the left.
  2. Vertical shift: 3 units upwards.

Would you like a detailed graph or further explanation of how to visualize the graph?

Here are 5 relative questions to expand on this information:

  1. How does changing the coefficient inside the absolute value function affect the graph?
  2. What happens to the graph if you subtract a value outside the absolute value function?
  3. Can you describe the graph of h(x)=x+1+3h(x) = -|x + 1| + 3?
  4. How does combining horizontal and vertical transformations change the graph of any function?
  5. What are other functions that have a V-shaped graph, and how do they compare to absolute value functions?

Tip: Always handle horizontal transformations (inside the function) first, as they impact the x-values directly, followed by vertical transformations (outside the function) that affect the y-values.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Transformations
Absolute Value Functions

Formulas

h(x) = |x + 1| + 3
f(x) = |x|

Theorems

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Suitable Grade Level

Grades 8-10