Graphs of absolute value functions

▶ Shifting absolute value graphs

The graph of y=|x-h|+k is the graph of y=|x| shifted h units to the right and k units up. See worked examples practicing this relationship.

⏱ 6:12 💾 6.2 MB License: cc-by-nc-sa Streaming via CDN

▶ Scaling & reflecting absolute value functions: equation

The graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from a description of the transformation performed on y=|x|.

⏱ 4:13 💾 3.8 MB License: cc-by-nc-sa Streaming via CDN

▶ Scaling & reflecting absolute value functions: graph

The graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from its graph.

⏱ 3:28 💾 5.4 MB License: cc-by-nc-sa Streaming via CDN

▶ Graphing absolute value functions

We can graph any absolute value equation of the form y=k|x-a|+h by thinking about function transformations (horizontal shifts, vertical shifts, reflections, and scalings).

⏱ 6:11 💾 9.4 MB License: cc-by-nc-sa Streaming via CDN

📄 Absolute value graphs review

General form of an absolute value equation:

$f(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}$

The variable $\goldD{a}$ tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables $\blueD h$ and $\blueD k$ tell us how far the graph shifts horizontally and vertically.

Some examples:

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Example problem 1

We're asked to graph:

$f(x)=|x-1|+5$

First, let's compare with the general form:

$f(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}$

The value of $\goldD a$ is $1$, so the graph opens upwards with a slope of $1$ (to the right of the vertex).

The value of $\blueD h$ is $1$ and the value of $\blueD k$ is $5$, so the vertex of the graph is shifted $1$ to the right and $5$ up from the origin.

Finally here's the graph of $y=f(x)$:

[[☃ image 6]]

Example problem 2

We're asked to graph:

$f(x)=-2|x|+4$

First, let's compare with the general form:

$f(x)=\goldD{a}|x-\blueD{h}|+\blueD{k}$

The value of $\goldD a$ is $-2$, so the graph opens downwards with a slope of $-2$ (to the right of the vertex).

The value of $\blueD h$ is $0$ and the value of $\blueD k$ is $4$, so the vertex of the graph is shifted $4$ up from the origin.

Finally here's the graph of $y=f(x)$:

[[☃ image 1]]

Want to learn more about absolute value graphs? Check out this video.

Want more practice? Check out this exercise.