Absolute Value Functions

Concept Explanation:

An absolute value function takes the form $f(x) = |x|$ , and it creates a V-shaped graph. The graph of $y = |x|$ has the following key features:

It is symmetric with respect to the y-axis.
The vertex of the graph is at $(0, 0)$ .
For $x \geq 0$ , $y = x$ .
For $x < 0$ , $y = -x$ .

Diagram:

The graph of $y = |x|$ :

Hard Questions:

Q1: Sketch the graph of $f(x) = |x - 2|$ and find its vertex.

Step-by-Step Solution:

The function $f(x) = |x - 2|$ is a horizontal translation of $y = |x|$ shifted 2 units to the right. The vertex is at $(2, 0)$ .

Answer:

The vertex is at $(2, 0)$ , and the graph is a V-shape.

Q2: Solve $|x - 5| = 7$ .

Step-by-Step Solution:

The absolute value equation gives two cases:

$x - 5 = 7$ or $x - 5 = -7$

Solving both:

$x = 12$ or $x = -2$

Answer:

The solutions are $x = 12$ and $x = -2$ .

Q3: Find the domain and range of $f(x) = |2x + 1|$ .

Step-by-Step Solution:

The domain of $f(x) = |2x + 1|$ is all real numbers because absolute value functions are defined for all $x$ . The range is $[0, \infty)$ because absolute value functions are non-negative.

Answer:

The domain is $(-\infty, \infty)$ and the range is $[0, \infty)$ .

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