Desmos for the SAT · Technique 4 of 15
Read a Parabola's Vertex, Roots, and Intercepts
Every "find the max, min, or zero" question turns into pointing at a graph.
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.
You almost never need to complete the square or factor by hand. Type the function into Desmos, click the feature the question is asking about, and read the coordinates it shows you.
Different question words point to different grey dots. Learn the map once (the reference below) and the question is telling you exactly where to click.
Nova
Parabola questions love to ask for the max, the min, or the zeros, and this move hands you all three. Type the function, click the point, read the number. The stuff people lose points on by hand becomes a one-click answer.
The algebra way vs the Desmos way
This technique sidesteps factoring or completing the square.
The algebra way
- Factor
x^2 - 4x - 5into(x - 5)(x + 1), so the roots arex = 5andx = -1. - For the vertex, find
x = -b/(2a) = 4/2 = 2, then plug back in to gety = -9. - Every factoring guess or sign slip along the way risks the point.
The Desmos way
- Type
y = x^2 - 4x - 5to graph the parabola. - Click the root, vertex, or intercept the question asks for and read the label.
The payoff: One graph and one click reads the exact coordinates, instead of factoring and vertex algebra where a single slip costs the question.
See the move, live
That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.
Quick reference
The question word points you to a dot
| The question asks for... | Click this grey 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 |
Before you start: set up Desmos
Convert decimals to fractions
Desmos shows decimals like 1.333. SAT choices are usually fractions. Click the round fraction-converter button next to a value to turn 1.333 into 4/3.
Turn on Projector Mode for clearer graphs
Desmos draws thin lines by default, which makes a crossing or intercept hard to pin down. Open the wrench icon (Graph Settings, top right) and turn on Projector Mode for thicker lines and larger labels, so the points you click are easier to hit exactly.
Widen the axis bounds when the graph looks empty
Desmos opens on a small window from -10 to 10 on both axes. If a crossing, vertex, or data point sits outside that box, the graph can look empty even when an answer exists. Open the wrench icon and widen the x and y bounds until the point you need comes into view.
Step by step
Nova
Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.
Type the function exactly as given, e.g. f(x) = x2 - 4x - 5.
Now try it on a real SAT question
The function f is defined by f(x) = x2 - 6x + 8.
What is the minimum value of f(x)?
Pick an answer to get instant coaching from Nova.
Minimum VALUE is the y-number at the bottom, not the x where it happens. Click the vertex dot and read the second coordinate off the label.
Click the lowest point on the parabola. Desmos labels it (3, -1). The question asks for the minimum value, which is the y-coordinate: -1.
In the app, drills like this adapt to your level, and your AI coach can walk you through any step you get stuck on.
Nova
Minimum VALUE is the y-number at the bottom, not the x where it happens. Click the vertex dot and read the second coordinate off the label.
Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.
Watch it
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.
- SAT trap answers are often exactly 0.5 off the nearest integer. If there is no grey dot sitting at a coordinate, it is not a real point of interest, so do not pick it. Click the dot Desmos actually drew and read that number.
- Desmos will not factor or simplify symbolically. It solves visually, so read the dots rather than expecting factored output.
- If the parabola opens off the screen and you cannot find the vertex or roots, type a couple of its coordinates into a table. That gives Desmos points to frame, so the Zoom Fit button appears and pulls the whole curve back into view.
Nova
Gut-check first though: if the equation hands you the answer, don't bother graphing. Vertex form already shows the vertex, and the sign of the leading coefficient already tells you max or min. Save Desmos for when the algebra would actually cost you time.
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 freeSAT 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.