Solve SAT Quadratics on Desmos: Zeros, Vertex, Fit

For most SAT quadratic questions, graph the function on Desmos and click it: the gray points show the zeros and the vertex. When the question gives you three points instead of an equation, put them in a table and let Desmos fit y₁ ~ ax₁² + bx₁ + c. When the question asks for an equivalent form, such as vertex form, algebra is usually quicker.

You can try each step below in the College Board version of Desmos on our Desmos on the SAT page. This guide gives you the rule, one worked example, one guided example and five practice problems with answers. Each answer was checked by hand.

The rule: what the graph of a quadratic tells you

A quadratic function has the form f(x) = ax² + bx + c. Its graph is a parabola. Three features answer most SAT questions about it:

Desmos marks these spots with gray points of interest. Select the curve or its expression line to see them, then hover over a point to read its coordinates. Click a point and its label stays on screen.

Worked example (I do): zeros and minimum from a graph

SAT-style question. The function f is defined by f(x) = x² - 6x - 16. What are the zeros of f, and what is the minimum value of f?

Step 1. Type the function on line 1.

1  y = x^2 - 6x - 16

Step 2. Read the zeros. Click the parabola. Two gray points sit on the x-axis, at (-2, 0) and (8, 0). The zeros are -2 and 8.

Step 3. Find the vertex. In the default window you may not see the bottom of the curve, because it sits at y = -25. Press the minus button at the top right of the graph to zoom out. A gray point appears at the lowest spot. Hover over it: (3, -25).

Step 4. Answer the question that was asked. The minimum value is the y-coordinate, so the answer is -25. The 3 only tells you where the minimum happens.

By-hand check. x² - 6x - 16 = (x - 8)(x + 2), so the zeros are 8 and -2. The vertex is halfway between them at x = (8 + (-2))/2 = 3, and f(3) = 9 - 18 - 16 = -25. Same answers. For more questions like this one, see our guide to finding a quadratic's minimum on Desmos.

Guided example (we do): build the quadratic from three points

SAT-style question. The total distance d, in meters, that an object travels is modeled by a quadratic function of time t, in seconds. The object has traveled 0 meters at t = 0, 50 meters at t = 10 and 200 meters at t = 20. How far has it traveled at t = 30?

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You don't have an equation yet, so there is nothing to graph. You have three points, and three points pin down one quadratic.

Step 1. Write the points. Time is x and distance is y: (0, 0), (10, 50) and (20, 200).

Step 2. Put them in a table. Add a table and type the points in its x₁ and y₁ columns.

Step 3. Fit the model. On a new line, type the quadratic with a tilde (~) where the equals sign would go, using the table's column names:

2  y_1 ~ ax_1^2 + bx_1 + c

The best fit is a = 0.5, b = 0 and c = 0. Desmos stores the fitted values, so later lines can use them.

Step 4 (your step). Turn the fit into a function and evaluate it at 30:

3  f(x) = ax^2 + bx + c
4  f(30)

Check your answer: d = 0.5t², so d(30) = 0.5 × 900 = 450 meters. By hand, c = 0 from the first point. The other two give 100a + 10b = 50 and 400a + 20b = 200. Subtract twice the first from the second: 200a = 100, so a = 0.5 and b = 0.

Tables with messy data and a line of best fit work the same way. Our SAT regression practice problems on Desmos cover those.

Your turn (you do): five SAT-style practice problems

Graph first, then check by hand. Answers are at the end of this section.

1. The function g is defined by g(x) = 2x² - 12x + 10. What is the minimum value of g?
2. A ball's height, in feet, t seconds after it is thrown is h(t) = -16t² + 48t + 4. What is the greatest height the ball reaches?
3. A quadratic function q passes through (0, 3), (1, 0) and (2, -1). What is q(5)?
4. What is the positive solution to x² + 3x = 28? (Graph y = x^2 + 3x and y = 28 on separate lines, then click the curve to see the gray intersection points.)
5. How many solutions does x² - 4x + 7 = 3 have?

Answers.

1. -8. The vertex is (3, -8). By hand: x = 12/4 = 3 and g(3) = 18 - 36 + 10 = -8.
2. 40 feet. The vertex is (1.5, 40). By hand: t = 48/32 = 1.5 and h(1.5) = -36 + 72 + 4 = 40.
3. 8. The fit gives a = 1, b = -4 and c = 3, so q(x) = x² - 4x + 3 and q(5) = 25 - 20 + 3 = 8.
4. 4. The graphs meet at x = -7 and x = 4. By hand: x² + 3x - 28 = (x + 7)(x - 4).
5. One. The equation becomes x² - 4x + 4 = 0, which is (x - 2)² = 0. On Desmos, y = x^2 - 4x + 7 touches y = 3 at the single point (2, 3).

If a question asks for the sum of the solutions rather than the solutions themselves, you can skip the graph. Our page on the sum of solutions formula shows how.

When algebra is faster

College Board's calculator guidance says some Math questions are better done without a calculator, even though one is allowed. For quadratics, that usually means questions about form. If you see f(x) = 3(x - 4)² + 7, you can read the vertex (4, 7) and the minimum value 7 straight from the equation. A graph adds nothing. The same goes for "which expression is equivalent" questions. Desmos can confirm that two expressions match, but rewriting by hand is usually quicker. Our guide to vertex form on Desmos covers those questions.

Mistakes to avoid

• Reporting the wrong coordinate. The vertex of y = x² - 6x - 16 is (3, -25). "Minimum value" means -25. "Where the minimum occurs" means x = 3.
• Using = in the regression line. The fit needs a tilde, and it needs the table's column names. Type y_1 ~ ax_1^2 + bx_1 + c, not y = ax^2 + bx + c.
• Stopping at the model. In the guided example, a = 0.5 is not the answer. The question asks for the distance at t = 30, which is 450.

Practice next

Start with the free Quadratic Functions lesson notes, then the Quadratic Equations lesson notes. For mixed questions, browse the SAT question bank.

Checked on September 26, 2026 against College Board's SAT Suite calculator policy, the College Board version of the Desmos graphing calculator, and the Desmos Help Center articles on regressions and points of interest. Every answer on this page was recomputed before publishing. The Test Advantage is independent of College Board and Desmos.