CalcHack Try it free

Vertex and maximum value: graph it, tap the turning point

An original SAT-style practice question on the maximum of a quadratic, worked step by step in Desmos — the same graphing calculator that's built into Bluebook — with the x = −b/(2a) method alongside.

Practice question

A model rocket is launched from a platform. Its height above the ground, in feet, t seconds after launch is modeled by h(t) = −16t² + 96t + 4. According to the model, what is the maximum height, in feet, the rocket reaches?

⚡ Desmos is the fastest way here The highest point of the parabola is the answer, and Desmos labels it the moment you tap it.
QuadraticsVertex / max-minMedium⏱ ~30 sec

Step by step

1

Graph the function, using x for t

Desmos graphs in x and y, so swap t for x. The y-values here run into the hundreds, so if the curve shoots off the screen, tap the zoom-fit button (the magnifying glass) to bring the top into view.

y=-16x^{2}+96x+4A parabola opening downward appears
1234564080120160 (3, 148) x = 3
What you see: an upside-down U that starts just above the origin, peaks in the middle, and comes back down near x = 6.
2

Tap the turning point

Tap the curve, then tap the grey point at the very top. Desmos labels the vertex with both coordinates.

(3,148)Highest point of the curve
What you see: a labelled point at (3, 148). The x-value is the time; the y-value is the height.
3

Pick the coordinate the question asked for

The question asks for the maximum height, which is the y-coordinate. The x-coordinate, 3, is when the rocket reaches it.

148Height in feet — done
Answer
148 feet

The vertex is at (3, 148): the rocket peaks 3 seconds after launch, at a height of 148 feet. Check: −16(3)² + 96(3) + 4 = −144 + 288 + 4 = 148 ✓.

⚠ Common trap

Answering 3 — that's the time of the maximum (the x-coordinate), not the maximum height. Flip it around too: if a question asks when the maximum happens, the answer is the x-coordinate, not 148. Answer choices often include both on purpose.

The algebra way, if you prefer

Axis of symmetry: for y = ax² + bx + c, the vertex sits at x = −b/(2a). Here a = −16 and b = 96, so x = −96 / (2 · −16) = 3. Plug that back in: h(3) = 148. Since a is negative, the parabola opens down and the vertex is a maximum.

That's about as fast as graphing once you know the formula — the extra step is remembering to substitute back in for the y-value.

More walkthroughs: sum of a quadratic's solutions · systems of linear equations · quadratic roots · circle equations · exponential growth · line of best fit · solving for x by graphing · linear inequalities · absolute value equations · Desmos cheat sheet · all walkthroughs

Stuck on a different practice question?

Snap a photo of any SAT-style math practice question and CalcHack walks you through the fastest route, in Desmos or by hand. Five free scans every day, no account needed to start.

Try it on your own question