Home›Blog›How to Find a Parabola's Vertex and Intercepts in Desmos

How to Find a Parabola's Vertex and Intercepts in Desmos

Last updated: October 6, 2026

Type the quadratic into Desmos and click the parabola. Desmos drops gray dots on the vertex, the y-intercept and both x-intercepts, and each one shows its exact coordinates when you click it. Your job is picking the right dot and the right coordinate for what the question asks.

The problem

Solve: x² − 2x − 3 = 0
Solution: x = −1 and x = 3

The same graph answers every other parabola question about this function. The parabola y = x² − 2x − 3 has its vertex at (1, −4) and crosses the y-axis at (0, −3).

What do you type?

One row: y = x^2 − 2x − 3

Desmos draws a parabola opening upward. Click anywhere on the curve and four gray dots appear:

• (−1, 0) and (3, 0): the x-intercepts, where y is zero
• (0, −3): the y-intercept
• (1, −4): the vertex, the lowest point on this parabola

Click a dot and its coordinates stay on screen. That's the whole method. You don't need to complete the square, use −b/2a, or rewrite anything in vertex form.

Which coordinate does the question want?

This is where points get lost. The dot is right; the number you copy from it is wrong.

• "What is the minimum value of f?" The y-coordinate of the vertex: −4
• "At what value of x does f reach its minimum?" The x-coordinate of the vertex: 1
• "What is the axis of symmetry?" The vertical line through the vertex: x = 1
• "What are the zeros of f?" The x-coordinates of the axis crossings: −1 and 3
• "What is the y-intercept?" The point (0, −3), or just −3 if they ask for the value

Before you type your answer, reread the last line of the question and say which coordinate it names.

What about word problems?

Projectile questions look like h(t) = −16t² + 64t + 5, the height of a ball t seconds after it's thrown. Replace t with x and type y = −16x^2 + 64x + 5.

The parabola opens downward, so the vertex is now a maximum. It's also far above the default window. Scroll out until you can see the top of the curve, then click it: (2, 69).

• "What is the maximum height?" 69, the y-coordinate
• "When does it reach its maximum height?" 2 seconds, the x-coordinate
• "When does it hit the ground?" The x-intercept on the right, where height is zero

Ignore the left x-intercept. It's at a negative time, before the ball was thrown.

What if the equation has a letter you don't know?

Some questions give you vertex form with a missing constant: "The graph of y = a(x − 1)² − 4 passes through (3, 0). What is a?"

Type y = a(x − 1)^2 − 4. Desmos offers to add a slider for a. Click it, then type the point (3, 0) on its own row. Drag the slider until the parabola runs through the point. Here that's a = 1, which is the parabola from the problem block.

If the slider skips past the value, click the number next to a and type answer choices in directly.

How do you check a vertex-form answer choice?

When the choices are four rewrites of the same function, type each one on a new row. The correct one lands exactly on top of your original parabola, and the wrong ones split off. For this problem, y = (x − 1)² − 4 overlaps perfectly. Its vertex, (1, −4), is the same gray dot.

Why vertex form shows (h, k) and factored form shows the zeros is covered in SAT Nonlinear Functions. For questions that only ask for the solutions, How to Solve SAT Quadratic Equations with Desmos covers counting crossings.

How do you practice this?

Take 10 parabola questions from practice tests: a few asking for a vertex, a few for intercepts, at least two word problems. For each one:

1. Graph it before you try any algebra, and scroll out until the vertex and both crossings are on screen
2. Click the dot the question points to and write down both coordinates
3. Circle the one coordinate the question actually asked for, then answer

When you miss one, check whether you clicked the wrong dot or read the wrong coordinate off the right dot. Those are different mistakes, and the second one is far more common.

HIROSCORE tracks nonlinear functions as their own skill, apart from the rest of Advanced Math, so a habit of copying the wrong coordinate shows up in your results instead of getting lost in an overall math score. The GPS for your SAT score.