Explain the mistake that is made. Describe a procedure for...

Explain the mistake that is made. Describe a procedure for graphing the function $f(x)=|3-x|+1$ Solution: a. Start with the function $f(x)=|x|$. b. Reflect the function about the $y$ -axis. c. Shift the function to the left three units. d. Shift the function up one unit. This is incorrect. What mistake was madc?

Explain the mistake that is made.
Describe a procedure for graphing the function $f(x)=|3-x|+1$
Solution:
a. Start with the function $f(x)=|x|$.
b. Reflect the function about the $y$ -axis.
c. Shift the function to the left three units.
d. Shift the function up one unit. This is incorrect. What mistake was madc?

Key Concepts

Absolute Value Function

The basic absolute value function, defined as f(x) = |x|, has a characteristic V-shape with its vertex at the origin. Understanding its structure is important when applying transformations, since its symmetry makes some alterations, like reflections, redundant or even misleading when not applied correctly.

Function Transformations

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. Correctly applying these operations in the proper sequence is essential for accurately graphing a transformed function. In this problem, misapplying a reflection led to an incorrect description of the transformation sequence.

Reflection

Reflection is a transformation that creates a mirror image of the function with respect to a given axis. However, for absolute value functions like f(x) = |x|, unnecessary reflections can complicate the process when the intended shift is simply a horizontal translation. In this case, reflecting f(x) = |x| about the y-axis is not needed to obtain f(x) = |3 - x| + 1.

Horizontal Translation

A horizontal translation shifts the graph left or right. For the function f(x) = |3-x| + 1, recognizing that it is equivalent to |x-3| + 1 means that the correct horizontal shift is three units to the right. The mistake was in interpreting the function transformation as requiring a shift to the left after an unnecessary reflection, rather than directly shifting the vertex to x = 3.

Transcript

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