Find a Parabola's Vertex & Roots on the SAT with Desmos
Quadratics show up everywhere on SAT Math. Desmos hands you the vertex, the roots, and the y-intercept without any factoring or the quadratic formula.
Step by step
- 1
Graph the quadratic as written
Type y = ax² + bx + c (e.g. y = x² − 6x + 5) on one line. Desmos draws the parabola instantly — no factoring.
- 2
Click the vertex
Click the lowest (or highest) point of the curve. Desmos labels the exact vertex (h, k) — that's your minimum or maximum value.
- 3
Click the x-intercepts for the roots
Click each point where the parabola crosses the x-axis. Those x-values are the solutions (roots) of the quadratic.
- 4
Read the y-intercept
Click where the curve crosses the y-axis to read the constant term / starting value.
- 5
Answer what's asked
Minimum value? Read the vertex's y. Roots? Read the x-intercepts. Axis of symmetry? It's the vertex's x.
Pro tip
A parabola opens up when a > 0 (it has a minimum) and down when a < 0 (a maximum). The vertex's x-coordinate is always the axis of symmetry, x = −b / 2a.
Try it yourself
Work the example right here in a live Desmos calculator — no Bluebook needed.
For f(x) = x² − 6x + 5, what is the minimum value of f ?
Graph the parabola and click the lowest point — that's the vertex (min value).
Loading interactive calculator…
Show the answer
Answer: −4
Graph it and click the lowest point: the vertex is (3, −4), so the minimum value is −4. (Algebra check: x = −b/2a = 3, and f(3) = 9 − 18 + 5 = −4.)
Put the trick to work on a real test
Full-length adaptive SAT mocks · real Bluebook-style timing · detailed score reports