Solve by Element List Regression

Determining Coefficients of a Polynomial on the SAT: Desmos Walkthrough

Build the parabola in vertex form with a slider for

a

, set

a

so it meets the condition, and let Desmos evaluate the expression you need.

This lesson uses sliders and direct evaluation in Desmos: put the unknown constant on a slider or a single expression line and let the calculator find the value instead of solving algebra by hand.

Take this lesson interactively

A real SAT-style question

In the xy-plane, a parabola has vertex $left parenthesis 9 comma negative 14 right parenthesis$ and intersects the x-axis at two points. If the equation of the parabola is written in the form $y = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants, which of the following could be the value of $a + b + c$ ?

$negative 23$

$negative 19$

$negative 14$

$negative 12$

How to solve it in Desmos, step by step

The idea

Use vertex form so the vertex (9, negative 14) is built in, add a slider for a, then set a to a positive value that makes the curve cross the x-axis twice and have Desmos read off a plus b plus c.

1
Step 1
Type the vertex form using the vertex (9, negative 14). Desmos pops up an Add Slider button for a, so tap it to create the a slider.
Type in Desmos
$y=a\left(x-9\right)^{2}-14$
2
Step 2
Set the a slider to a positive number so the parabola opens upward and crosses the x-axis at two points, which is the condition the question requires. A negative a would open downward and miss the axis. Set a to 1 over 32 by typing this line.
Type in Desmos
$a=\frac{1}{32}$
3
Step 3
Add this line to display a plus b plus c. Expanding a times (x minus 9) squared minus 14 gives a, then negative 18 a, then 81 a minus 14, which sum to 64 a minus 14. With a at 1 over 32 Desmos shows the value negative 12, matching choice D. Because a must stay positive, 64 a minus 14 is always greater than negative 14, so the other choices are impossible.
Type in Desmos
$64a-14$

The answer is D.

$negative 12$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

In the

xy

-plane, a parabola has vertex

(6, -20)

and intersects the

x

-axis at two points. If the equation of the parabola is written in the form

y = ax^2 + bx + c

, where

a

b

, and

c

are constants, which of the following could be the value of

a + b + c

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson

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