Graph PlottersPlot the V-shaped absolute value graph and see its vertex and axis update as you change the equation.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
The absolute value graph is a V-shaped parent curve. Its vertex and direction are controlled by the coefficients around |x|.
HOW TO USE IT
The parent absolute value function y = |x| forms a V-shaped graph with vertex at the origin. In y = A|Bx + C| + D, A changes the vertical direction and scale, B changes horizontal scale, C moves the vertex horizontally, and D moves it vertically.
When B is nonzero, the vertex occurs at x = −C/B and y = D. A positive A creates a minimum; a negative A creates a maximum. The two straight branches make absolute value useful for studying symmetry, distance from zero, and piecewise definitions.
Change one parameter at a time, try a negative scale, and use the vertex card to confirm the movement. The coordinate cursor helps inspect the two linear branches and compare their slopes.
The absolute value function measures distance from zero, so y = |x| has two straight branches meeting at a vertex. In y = A|Bx + C| + D, the vertex is (−C/B, D) when B is nonzero. A positive A opens the V upward and a negative A opens it downward. The magnitude of A changes branch steepness, while B changes horizontal scale. This makes absolute value a useful bridge between equations, distance, symmetry, and piecewise linear graphs.
For y = −2|x − 1| + 3, the vertex is (1, 3), the graph opens downward, and the branches have slopes 2 and −2. Plot it, confirm the vertex marker, then test x = 0 and x = 2; both points have y = 1 because the graph is symmetric about x = 1. Use this symmetry check whenever you are unsure whether a horizontal shift was entered with the correct sign.
For y = |x − 3| + 2, the vertex is (3, 2), the graph opens upward, and the range is y ≥ 2. The points (2, 3) and (4, 3) show symmetry around x = 3. If you change the coefficient to −2, the vertex stays fixed but the graph opens downward and becomes steeper. These simple test points make the transformation visible.
The expression |x − h| measures the distance between x and h, so the vertex’s x-coordinate is h. This is why an absolute value graph is symmetric and why it naturally appears in distance, tolerance, and error models. The graph calculator lets you see the same idea as an equation, a V-shape, and a pair of linear branches.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
The parent graph y = |x| has vertex (0, 0).
It reflects the absolute value graph across its vertex and makes the V open downward.
For A|Bx + C| + D with B nonzero, the vertex is (−C/B, D).
Absolute value changes negative inputs into positive distances, creating one line on each side of the vertex.
Set the inside equal to zero: x = −C/B when B is nonzero. The y-coordinate is D.
A negative A reflects the V-shape across its horizontal vertex level, turning a minimum into a maximum.