Desmos for the SAT · Technique 5 of 23

Read a Parabola's Vertex, Roots, and Intercepts

Every "find the max, min, or zero" question turns into pointing at a graph.

SolvesParabola vertexMax or minRoots and zeros

Watch the lesson

The whole move worked on one real SAT question, narrated.

0:00 / 0:00

A second example, a downward parabola: click the vertex, then read the number the question names.

Quadratic and other nonlinear function questions usually ask about one specific feature: the vertex (max or min), the zeros (x-intercepts), or the y-intercept.

For most of these, the graph answers faster than factoring or completing the square by hand. Type the function into Desmos, click the feature the question names, and read the coordinates it shows you.

Different question words point to different gray dots. Learn the map once (the reference below) and the question is telling you exactly where to click.

Nova

Parabola questions ask for the max, the min, or the zeros, and this move gives you all three. Type the function, click the point, read the number. The parts that are slow by hand become one click.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps the vertex formula, or completing the square.

The question

$y = x^{2} - 4x - 5$

What is the minimum value of f(x)?

The algebra way

Find the x-coordinate of the vertex: x = -b/(2a) = 4/2 = 2.
Substitute it back: f(2) = 4 - 8 - 5 = -9. A sign slip in either step lands on one of the wrong choices.

The Desmos way

Type y = x^2 - 4x - 5 and select it: dots appear on the roots, the vertex and the y-intercept.
Click the bottom of the curve and read the y-coordinate: -9.

The payoff: One graph and one click reads the exact vertex, instead of a formula and a substitution where a single slip costs the question.

Quick reference

The question word points you to a dot

The question asks for...	Click this gray dot
Roots, zeros, solutions, x-intercepts	The dots on the x-axis
Maximum, minimum, or extremum	The dot at the top or bottom (the vertex)
Starting value, or the value at $x = 0$	The dot on the y-axis

Step by step

The whole move, all 4 steps

Type the function on its own line, exactly as given.
Click inside the equation box to reveal the gray dots. Desmos marks the roots, the vertex and the intercepts at once.
Let the question word pick the dot: roots and zeros sit on the x-axis, a maximum or minimum is the vertex, a starting value is on the y-axis.
Click that dot and read the coordinate the question asks for. A “value” is the y-number, not the x where it happens.

The question

The function f is defined by $f(x)=x^{2}-4x-5$ .

What is the minimum value of f(x)?

Step 1 of 5

Type the function as y = x² - 4x - 5, exactly as given, with y standing in for f(x).

In the calculator: y=x^2-4x-5

Now try it on a real SAT question

Head or scratch paper. Vertex form already shows the vertex at $(3, -4)$ , so the minimum value is sitting in the equation. This is the case the lesson warns about: graphing it is slower than reading it.

The function $f(x) = (x - 3)^{2} - 4$ has a minimum value.

What is the minimum value of f(x)?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

If you only need to know whether the vertex is a max or a min, look at the leading coefficient in the equation: positive means min, negative means max. No graphing required.
If the function is already given in vertex form, a(x - h)² + k, the vertex is just (h, k). Reading it off the equation is faster than opening the calculator.
Desmos will not factor or simplify symbolically. It solves visually, so read the dots rather than expecting factored output.

If the calculator does something you did not expect

If there is no gray dot at a coordinate, Desmos did not mark a point there. Click the dot Desmos drew and read its label rather than estimating from the gridlines.
If the parabola runs off the screen and you cannot see the vertex or the roots, press the minus button to zoom out, or open the wrench icon and widen the y-axis bounds until the whole curve is in view.

All Desmos techniques

Put this move into real practice

"Read a Parabola's Vertex, Roots, and Intercepts" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.