How to find the vertex and zeros with Desmos on the SAT
Updated
Questions about zeros, x-intercepts, and maximum or minimum values are some of the most common on SAT math, and some of the easiest to do with Desmos. Desmos marks those points for you. You click, read the coordinates, and answer.
This guide covers the method, four SAT-style examples (a minimum, a word problem with a maximum, a "how many solutions" question, and a cubic), and the mix-ups that trip students up.
The method in three steps
- 1Type the function, like . You can leave off , or write if the question names the function.
- 2Click the curve. Gray points appear at its zeros (where it crosses the -axis), its vertex or other turning points, and its -intercept.
- 3Click a gray point to see its coordinates.
Then match the question to the right number: a zero is an -value where ; a maximum or minimum value is the -coordinate of a turning point.
Example 1: the minimum value
Read the right coordinate
Graph and click it. Desmos marks the vertex at and the zeros at and .
The minimum value is the lowest -value: . The answer is not , which is where the minimum happens.
Answer:
Example 2: a word problem with a maximum
Highest point, then the ground
Type it with in place of : . The graph is off screen at first, so zoom out until you see the top of the arc. Click the curve.
The vertex is : a maximum height of 43 feet, 1.5 seconds after the kick. The positive zero is , so the ball lands after about 3.1 seconds. Ignore the negative zero: time can't be negative.
Answer: 43 feet; about 3.1 seconds
Example 3: how many real solutions?
Count the zeros
- A. Zero
- B. Exactly one
- C. Exactly two
- D. Infinitely many
The solutions of are the zeros of . Graph and click it: the only gray points are the vertex and the -intercept. The parabola sits entirely above the -axis.
No crossings, so no real solutions. The algebra agrees: the discriminant is , which is negative.
Answer: Zero
Example 4: zeros of a cubic
Add up the solutions
Graph and click it. Desmos marks three zeros, plus the two turning points. The zeros are at , , and .
The sum is . Don't miss the zero at the origin: it's easy to overlook because it sits on the -axis.
Answer:
Mistakes that cost points
- Mixing up where and what. "The minimum value" is a -value. "Where the minimum occurs" or "the value of " is an -value.
- Clicking the wrong graph. Gray points only appear for the curve you click. With two functions on screen, click the one the question is about.
- Not zooming out. If you see no vertex, the graph's interesting part is off screen. Word problems with feet or dollars often need a big window.
- Answering with a long decimal. If Desmos shows and the question asks for the nearest tenth, round: .
Practice this move
Practice on real SAT-style problems
Intersect Prep puts the same Desmos calculator beside 33 SAT-style problems that use this move, with a step-by-step strategy for each one. Create a free account to try a full lesson with the calculator; a pass unlocks the rest.
Questions
- How do you find the vertex in Desmos?
- Graph the function and click the curve. Desmos marks the vertex (and any other turning point) with a gray point; click it to see its coordinates.
- How do you find zeros in Desmos?
- Graph the function and click it. The gray points on the x-axis are the zeros, also called roots, solutions, or x-intercepts.
- Does this work for functions that aren't parabolas?
- Yes. Desmos marks zeros and turning points for cubics, absolute value graphs, and most other functions on the SAT.
- What if Desmos shows no zeros?
- Then the graph never crosses the x-axis on screen. Zoom out to be sure. If it still doesn't, the equation has no real solutions.