Absolute value functions make graphs that form a clear V shape, which makes them easy to recognize and useful for modeling distance from a starting point. The parent function y = |x| has its vertex at the origin and is symmetric about the y-axis. Learning to graph these functions helps students connect equations, tables, and geometric transformations on the coordinate plane.
Most absolute value graphs can be written in the form y = a|x - h| + k, where the vertex is (h, k). The values h and k shift the V left, right, up, or down, while a changes the steepness and whether the graph opens upward or downward. By identifying the vertex, direction, and slope of each arm, students can sketch accurate graphs without making a full table of values.
Key Facts
- Parent function: y = |x|
- Vertex form: y = a|x - h| + k
- The vertex of y = a|x - h| + k is (h, k).
- If a > 0, the V opens upward; if a < 0, the V opens downward.
- The graph of y = |x| is symmetric about the y-axis.
- A larger |a| makes the V narrower, while 0 < |a| < 1 makes the V wider.
Vocabulary
- Absolute value
- The absolute value of a number is its distance from 0 on the number line.
- Parent function
- A parent function is the simplest form of a function family, such as y = |x| for absolute value functions.
- Vertex
- The vertex is the corner point of an absolute value graph where the two line segments meet.
- Transformation
- A transformation is a change to a graph, such as a shift, stretch, compression, or reflection.
- Axis of symmetry
- The axis of symmetry is the vertical line that divides an absolute value graph into two matching halves.
Common Mistakes to Avoid
- Treating y = |x - h| as a shift left by h is wrong because x - h shifts the graph right when h is positive.
- Forgetting that the vertex is (h, k) in y = a|x - h| + k is wrong because the sign inside the absolute value is opposite of what it may first appear.
- Using only one side of the V to graph is wrong because an absolute value graph has two linear arms that mirror each other around the axis of symmetry.
- Assuming a negative a moves the graph down is wrong because a negative value of a reflects the V across a horizontal line through the vertex.
Practice Questions
- 1 Graph y = |x - 3| + 2. Identify the vertex and axis of symmetry.
- 2 For y = -2|x + 1| + 4, find the vertex, state whether the graph opens up or down, and calculate y when x = 3.
- 3 Explain how the graph of y = 0.5|x - 2| - 3 is related to the graph of y = |x|, including shifts and changes in width.