Problems 134–137 are based on material learned earlier in...

Problems 134–137 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the function that is finally graphed if all of the following transformations are applied to the graph of $y=|x|$ (a) Shift left 3 units. (b) Reflect about the $x$ -axis. (c) Shift down 4 units.

Problems 134–137 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Write the function that is finally graphed if all of the following transformations are applied to the graph of $y=|x|$
(a) Shift left 3 units. (b) Reflect about the $x$ -axis. (c) Shift down 4 units.

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Key Concepts

Vertical Translation

Vertical translation involves adding or subtracting a constant to the function, shifting the graph up or down. For instance, subtracting 4 from the function moves the graph 4 units downward.

Reflection about the x-axis

Reflecting a function about the x-axis involves multiplying the function by -1, which inverts all the y-values of the graph. This transformation flips the graph vertically.

Transformations of Functions

This concept involves altering the original function's graph using various operations such as translations (shifts), reflections, stretches, and compressions. Understanding these transformations helps in predicting how the graph changes when the function's equation is modified.

Absolute Value Functions

Absolute value functions, exemplified by y = |x|, produce a V-shaped graph that is symmetric about the y-axis. They are often used to illustrate how transformations affect key points and the overall shape of the graph.

Horizontal Translation

A horizontal translation occurs when the input variable in a function is replaced by an expression of the form x + h or x - h, resulting in a shift of the graph left or right. For example, replacing x with x + 3 shifts the graph 3 units to the left.

Transcript

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