Playground: y = −a|x − h| + k

Move the sliders, or drag the dots on the graph: the big dot is the vertex (h, k) and the small dot changes a.

Hover over the graph to trace points.

Controls

Steepness and direction. The minus sign in front means a positive a flips the V upside down.
Left/right shift. Careful: x − 3 moves it right 3.
Up/down shift.

What the numbers are telling you

    📏 Domain & Range

    Domain = every x-value the graph uses. Squash the graph flat onto the x-axis to get the amber bar.
    Range = every y-value the graph reaches. Squash it flat onto the y-axis to get the teal bar.

    x from to

    📝 Description of the graph

    🎵 The V Song (tune: Twinkle, Twinkle, Little Star)

    Click a verse title to play just that verse. Turn off Melody to sing it yourself over the bass (karaoke mode).

    Fill-in sentence and chant

    "The graph of y = ___ is the graph of y = |x| reflected over the x-axis (only if a is positive), vertically stretched / compressed by a factor of ___, and translated ___ units right/left and ___ units up/down."

    Chant (We Will Rock You beat: stomp, stomp, clap):
    Y (stomp stomp) equals (clap) · Minus a (stomp stomp) bars (clap) · X minus h (stomp stomp) plus k! (clap) · Corner's at h, k! (stomp stomp clap)

    "Bigger a" means bigger size, ignoring the sign (−3 counts as bigger than 1). "Fractions" means anything between −1 and 1.

    🎯 Challenge: match the red dashed graph

    Press "New challenge" to start.

    📈 Graph your own function

    Use x, abs(), *, ^, sqrt(), sin()… You can also use the slider values a, h, k.

    Try:

    Table of values

    Highlighted row is the vertex. Notice the y-values are the same on both sides: the graph is symmetric.

    📘 Quick cheat sheet
    • Start with the parent graph y = |x|: a V with its corner at (0, 0) and slopes 1 and −1.
    • a stretches it. Bigger |a| means steeper and narrower. |a| between 0 and 1 means flatter and wider.
    • The minus sign in −a reflects the graph over the x-axis. With a > 0 the graph opens down (∧). If a is negative, the two minuses cancel and it opens up (V).
    • h moves it right (h > 0) or left (h < 0). The sign looks backwards: |x − 2| goes right 2 and |x + 2| goes left 2, because x + 2 = x − (−2).
    • k moves it up (k > 0) or down (k < 0).
    • The vertex (corner) is always at (h, k).
    • The right side has slope −a and the left side has slope +a.
    • Order to graph by hand: plot (h, k), then from there go 1 right and −a up/down, then mirror that on the left.
    • Domain is all real numbers, because any x can go inside |x − h|.
    • Range starts at the vertex height k: y ≤ k if a > 0 (opens down), y ≥ k if a < 0 (opens up).
    • Restricted domain? Find the range by checking the y-values at both ends and at the vertex (if the vertex is inside).