Absolute-value function
Graph and analyze absolute-value functions in Grade 9 Algebra. Identify the vertex, axis of symmetry, and V-shape from the equation f(x) = |x - h| + k.
Key Concepts
Property The absolute value parent function is $f(x) = |x|$. The graph forms a 'V' shape with a vertex at $(0, 0)$ and an axis of symmetry at $x = 0$. Explanation Think of absolute value as a 'distance machine.' It tells you how far a number is from zero, which is always positive. That's why the graph bounces back up from the x axis, creating its signature 'V' shape! Examples For example, both $x = 3$ and $x = 3$ result in $y = 3$, since the distance from zero is the same. The vertex, or 'corner', is at $(0,0).$.
Common Questions
What is an absolute-value function and what does its graph look like?
An absolute-value function has the form f(x) = a|x - h| + k, producing a V-shaped graph. The vertex is at (h, k), and the arms open upward when a > 0 and downward when a < 0.
How do you find the vertex of an absolute-value function?
The vertex is the point (h, k) in the equation f(x) = a|x - h| + k. Set the expression inside the absolute value equal to zero to find h, and k is the constant added outside. The vertex is the minimum or maximum depending on the sign of a.
What is the axis of symmetry of an absolute-value graph?
The axis of symmetry is the vertical line x = h passing through the vertex of the V-shape. The graph is a perfect mirror image on both sides of this line, making the absolute-value function symmetric about it.