Graphing Absolute Value Functions with Transformations

Math Problem Statement

Graph the functions y = |x|, y = -|x|, y = -4|x|, and y = -4|x - 3| on the same screen and identify which graph corresponds to each function.

To determine which graph matches each function, let's understand the transformations:

$y = |x|$ is a V-shaped graph opening upwards.
$y = -|x|$ is a V-shaped graph opening downwards.
$y = -4|x|$ is a vertically compressed and downward-opening V due to the negative and the factor 4.
$y = -4|x - 3|$ is similar to the previous, but horizontally shifted 3 units to the right.

Now, let's analyze the given graphs:

The graph in the first option (top-left) matches:
- (a) $y = |x|$ as the blue graph opening upwards.
- (b) $y = -|x|$ as the red graph opening downwards.
- (c) $y = -4|x|$ as the purple steeper graph opening downwards.
- (d) $y = -4|x - 3|$ as the green graph showing a rightward shift.

So, the first option (top-left) is the correct match.

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Absolute Value Graphs
Transformations of Functions
Reflections
Vertical Stretching and Compression
Horizontal Shifts

y = a|x - h| + k (general form of absolute value functions)

Transformations of functions theorem: Effects of a, h, and k in y = a|x - h| + k

Grades 9-11

Graphing Absolute Value Functions with Transformations: f(x) = |x|, g(x) = |4x|, h(x) = -4|x + 4| - 5

Graphing the Absolute Value Function y = -|x - 3|

Identify Transformations of Absolute Value Graphs: Reflections and Shifts

Sketching the Graph of f(x) = -2|x + 1| + 4

Graphing Absolute Value Functions: |x| + 3x and 2|x| - 4