Graphing f(x)=a|x-h|+k Using Vertex and Slope
Graphing f(x) = a|x - h| + k in Algebra 1 (California Reveal Math, Grade 9) uses two key features: the vertex at (h, k) and the slope |a| for the arms. Start at the vertex, then move right one unit and up |a| units (for the right arm), and left one unit and up |a| units (for the left arm). If a is negative, the arms point downward instead of upward. For example, f(x) = 2|x - 3| + 1 has vertex (3, 1) and each arm rises 2 units per horizontal unit. This graphing method makes absolute value function transformations systematic and precise.
Key Concepts
To graph $f(x) = a|x h| + k$, start at the vertex $(h, k)$ and use the slope $|a|$ to plot points on each side of the vertex. For $x h$, rise $|a|$ units for every 1 unit to the right. For $x < h$, rise $|a|$ units for every 1 unit to the left (mirror image). If $a < 0$, the V shape opens downward , so the arms fall away from the vertex instead of rising.
$$\text{Vertex: } (h,\, k) \qquad \text{Slope of right arm: } a \qquad \text{Slope of left arm: } a$$.
Common Questions
How do you graph f(x) = a|x - h| + k?
Plot the vertex at (h, k). From the vertex, move right 1 unit and up |a| units to plot a point on the right arm. Mirror this to the left. If a < 0, the arms go downward.
What does the vertex of an absolute value function represent?
The vertex (h, k) is the turning point of the V-shaped graph — the minimum if a > 0 or maximum if a < 0.
What does the value of a determine in the graph?
The absolute value of a (|a|) determines the slope of the arms. A larger |a| makes steeper arms; smaller |a| makes flatter arms. Negative a flips the graph downward.
What are the transformations represented by h and k?
h represents a horizontal shift: the vertex moves right h units (or left if h is negative). k represents a vertical shift: the vertex moves up k units (or down if negative).
Where is graphing absolute value functions covered in California Reveal Math Algebra 1?
This skill is taught in California Reveal Math, Algebra 1, as part of Grade 9 function transformations and absolute value functions.
How do you find the x-intercepts of f(x) = a|x - h| + k?
Set f(x) = 0 and solve a|x - h| + k = 0. Isolate the absolute value: |x - h| = -k/a. If -k/a ≥ 0, two solutions exist; if equal to 0, one; if negative, none.
What does a V-shaped graph with the vertex above the x-axis tell you about x-intercepts?
If a > 0 and k > 0, the vertex is above the x-axis and the arms open upward, meaning the function has NO x-intercepts.