Topic 3

Calculator Strategy (Desmos on the Digital SAT)

The Digital SAT provides a built-in Desmos graphing calculator for every math question in both modules. You can also bring your own approved calculator, but the Desmos tool is powerful and worth mastering because it can solve many problems faster than algebra.

Key Desmos capabilities for the SAT:

Graphing equations: Type any equation (e.g.,

y = 2x^2 - 3x + 1

) and Desmos plots it instantly. You can graph two equations simultaneously and visually identify intersection points. Click on an intersection point and Desmos displays its coordinates.

Solving equations: To solve

2x + 5 = 13

, graph

y = 2x + 5

and

y = 13

and find the intersection. For a quadratic like

x^2 - 5x + 6 = 0

, graph

y = x^2 - 5x + 6

and look for the

x

-intercepts (where the curve crosses

y = 0

). Click on the intercepts and Desmos gives you exact values.

Systems of equations: Graph both equations and find the intersection point(s). This works for linear-linear, linear-quadratic, and even two quadratic equations.

Regression and data: If a problem gives you a table of data, you can enter the points and use Desmos to fit a line or curve.

Sliders: When an equation has a parameter (like

y = ax + 2

where

a

is unknown), Desmos offers a slider for

a

. You can drag the slider to see how the graph changes. This is useful for ``for what value of

k

does...'' problems.

When to use Desmos vs.\ algebra:

Use Desmos when:

You need to find intersection points of two graphs.
You need roots of a quadratic or higher-degree polynomial.
A problem asks how many solutions a system has (graph both and count intersections).
You want to check your algebraic work quickly.
The algebra is messy and a visual approach is cleaner.

Use algebra when:

The problem asks for an exact symbolic answer (e.g., ``in terms of $k$ '').
The arithmetic is straightforward (solving $3x = 15$ does not need a graph).
The problem involves simplifying an expression rather than solving an equation.

Common Desmos pitfalls:

Desmos sometimes displays approximate decimal coordinates for intersection points. If the SAT expects an exact answer like

\dfrac{7}{3}

, you need to recognize that

2.333...

means

\dfrac{7}{3}

Zooming is important. If you graph a function and do not see an intersection or root, try zooming out. The default window may not show the part of the graph you need.

Core Skills

Use Desmos to find roots of equations by graphing and identifying $x$ -intercepts.
Solve systems of equations by graphing both equations and reading intersection coordinates.
Use sliders to determine parameter values that produce specific graph behaviors (tangency, number of solutions).
Convert Desmos decimal output to exact fractions or radicals when the problem requires it.
Decide quickly whether Desmos or algebra is the faster approach for a given problem.

Example 1: Finding Roots with Desmos

Solve

x^2 - 7x + 10 = 0

. Desmos approach: Graph

y = x^2 - 7x + 10

. The parabola crosses the

x

-axis at

x = 2

and

x = 5

. Click each intercept to confirm. Algebra check:

(x - 2)(x - 5) = 0

\boxed{x = 2 \text{ and } x = 5}

Example 2: Solving a System by Graphing

Find the intersection of

y = x^2 - 1

and

y = 2x + 2

. Desmos approach: Graph both equations. The graphs intersect at two points. Click to read coordinates:

(-1, 0)

and

(3, 8)

. Algebra check:

x^2 - 1 = 2x + 2 \implies x^2 - 2x - 3 = 0 \implies (x-3)(x+1) = 0

\boxed{(-1, 0) \text{ and } (3, 8)}

Example 3: Using a Slider for a Parameter

For what value of

k

is the line

y = 3x + k

tangent to the parabola

y = x^2

? Desmos approach: Graph

y = x^2

and

y = 3x + k

. Add a slider for

k

. Drag the slider until the line just touches the parabola at exactly one point. The slider reads

k = -2.25

, which is

-\dfrac{9}{4}

. Algebra check: Set

x^2 = 3x + k

, giving

x^2 - 3x - k = 0

. Discriminant

= 0

9 + 4k = 0 \implies k = -\dfrac{9}{4}

\boxed{-\dfrac{9}{4}}

Example 4: Converting Desmos Decimals to Exact Answers

A Desmos intersection reads

(1.6667, 4.3333)

. The SAT asks for the coordinates as fractions. What are they? Step 1: Recognize the repeating decimals.

1.6667 \approx 1.\overline{6} = \dfrac{5}{3}

and

4.3333 \approx 4.\overline{3} = \dfrac{13}{3}

\boxed{\left(\dfrac{5}{3},\, \dfrac{13}{3}\right)}

Key Takeaways

Desmos is fastest for ``find the intersection'' and ``how many solutions'' problems. Graph both sides and read the answer directly.
For ``find the value of $k$ '' problems, use a slider. Drag until the desired condition (tangency, specific intercept, etc.) is met.
Always be ready to convert Desmos decimals to exact values. Common conversions: $0.\overline{3} = \frac{1}{3}$ , $0.\overline{6} = \frac{2}{3}$ , $0.25 = \frac{1}{4}$ , $0.1\overline{6} = \frac{1}{6}$ .
Desmos is a checking tool even when you solve algebraically. A 10-second graph confirms your answer and catches errors.