SAT® · Test Mechanics: The Tools and the Traps
Desmos: Zeros, Vertices, and Key Points
Desmos turns algebra questions into graph-reading questions: type an equation, click the curve, and it labels zeros, vertices, and y-intercepts for you.
Finding where a quadratic equals zero is really asking where its graph crosses the x-axis. The x-axis is just the line y = 0, so a zero is simply a crossing point between the curve and that line. Desmos finds crossings between any two curves you type in, which means you don't need to factor or use the quadratic formula at all. Type the equation, click directly on the curve where it looks interesting, and Desmos pops up a label with the exact coordinates of that point: a zero where it crosses the x-axis, the vertex at the top or bottom of the parabola, or the y-intercept where it crosses the y-axis.
This trick extends past single equations. If a question asks where x2 - 5x + 3 equals 2x - 4, don't move everything to one side and factor. Graph y = x2 - 5x + 3 and y = 2x - 4 as two separate curves. Wherever they physically cross on screen, the x-coordinates are exactly the solutions to the original equation, because that's where both sides give the same output. Click each crossing point to read off its coordinates directly.
This connects to the rest of the toolkit because it replaces algebra with observation: instead of solving, you're reading labeled points off a picture. The catch is that a picture has limited resolution. Two curves that look like they touch at one point on screen might actually cross twice very close together, or might not touch at all. That's where the discriminant from the formula card still matters, because it settles exactly how many real roots exist when the screen can't.
Key terms
- Zero
- An x-value where a graph crosses or touches the x-axis, meaning y equals 0 there.
- Vertex
- The highest or lowest point on a parabola, marking where it turns direction.
- Discriminant
- The expression b2 - 4ac from the quadratic formula, which tells you how many real roots an equation has.
- Intersection point
- A point where two graphed curves share the same x and y values, giving the solution to setting them equal.
Watch it explained
- How it worksHow does clicking a curve in Desmos hand you the zeros, the vertex and the y-intercept?
- Worked exampleFind the zeros and the vertex of y = 2x2 - 8x + 6 by graphing it instead of factoring.
- Worked exampleFind where x2 - 5x + 3 equals 2x - 4 by graphing both sides as two separate curves.
- Common mistakeWhy doesn't a curve that looks like it touches zero on screen prove the root is exactly there?
Check yourself
End of topic.