What It Targets

• Systems of Equations
• Function Evaluation
• Quadratics
• Graph Transformations
• Intersections
• X-Intercepts
• Discriminant Questions
• SAT Calculator Strategy

The Core Idea

Many SAT Math questions can be solved faster by graphing than by doing algebra. The Digital SAT includes Desmos for a reason. Your goal is not to avoid learning math. Your goal is to recognize when the graph can give you the answer faster than solving by hand. Think of Desmos as a calculator for graphs.

When Should I Use Desmos?

Look for questions that ask about:

• Intersections
• Solutions to systems
• X-intercepts
• Y-intercepts
• Maximums or minimums
• Vertices
• Function values
• Shifts and transformations
• Number of solutions

These are often graphable in under 30 seconds.

The Three Most Useful Desmos Moves

1. Graph Both Equations

If the SAT gives you two equations:

Graph both.

The intersection point is often the answer.

2. Type Function Notation Directly

Example:

f(x) = 2^x + 3

Then simply type:

f(4)

Desmos will calculate it instantly.

3. Use Sliders

If a variable such as a, b, or k appears in the equation:

Create a slider.

Move the slider and watch the graph change.

This is especially useful for:

• discriminant questions
• transformations
• optimization questions

Recognition Checklist

Before solving, ask:

□ Can I graph this?

□ Is the question asking for an intersection?

□ Is it asking for a maximum or minimum?

□ Is it asking for an intercept?

□ Is it asking for the number of solutions?

□ Is there a parameter I can control with a slider?

If yes, Desmos is probably faster than algebra.

Guided Practice

Open Desmos Graphing Calculator

Question 1

If 3x - 2y = 12 and x + 4y = 18, what is the value of y?

Answer: 3

Desmos Method

Type both equations into Desmos:

3x - 2y = 12

x + 4y = 18

Click the intersection point.

You should see:

(6, 3)

The question asks for the y-value.

Answer:

3

Question 2

A quadratic function is defined by

f(x) = -2(x - 3)² + 8

What is the maximum value of f(x)?

Answer: 8

Desmos Method

Graph the function.

Look at the vertex.

The highest point is:

(3, 8)

The maximum value is the y-coordinate.

Answer:

8

Question 3

Find the y-coordinate of the intersection for the system:

y = x² - 4x + 5

y = 2x - 3

Answer: 1 or 5

Desmos Method

Graph both equations.

The graphs intersect twice:

(2, 1)

and

(4, 5)

Both y-values are intersections.

Answer:

1 or 5

Question 4

If

g(x) = 3x - 7

and

g(k) = 14

what is the value of k?

Answer: 7

Desmos Method

Graph:

y = 3x - 7

Then graph:

y = 14

The graphs intersect at:

(7, 14)

Answer:

7

Question 5

How many solutions does the system

y = 3x + 5

and

2y - 6x = 10

have?

Answer: Infinitely Many

Desmos Method

Graph both equations.

The lines overlap perfectly.

A system with the same line has infinitely many solutions.

Question 6

What is the x-intercept of the line

4x - 7y = 28

Answer: 7

Desmos Method

Graph the equation.

Locate where the graph crosses the x-axis.

The point is:

(7, 0)

Answer:

7

Question 7

For what value of b will the equation

x² + bx + 16 = 0

have exactly one real solution?

Answer: 8 or -8

Desmos Method

Graph:

y = x² + bx + 16

Create a slider for b.

Move the slider until the parabola touches the x-axis exactly once.

This occurs when:

b = 8

or

b = -8

Question 8

If

f(x) = 2^x + 3

what is the value of f(4)?

Answer: 19

Desmos Method

Type:

f(x) = 2^x + 3

On the next line type:

f(4)

Desmos instantly returns:

19

Question 9

Find the intersection of

1.3x - 0.6y = -0.7

and

6.5x - 1.5y = -0.5

What is the y-coordinate?

Answer: 2

Desmos Method

Graph both equations.

Zoom in if necessary.

The intersection point is:

(0.5, 2)

Answer:

2

Question 10

The graph of

y = x²

is shifted 3 units left and 5 units up.

Write the resulting equation.

Answer

y = (x + 3)² + 5

Desmos Method

Graph:

y = x²

Then graph:

y = (x + 3)² + 5

Verify that the parabola moved:

• 3 units left
• 5 units up

Speed Drill

Try these without doing any algebra first.

Use Desmos immediately.

1

Find the x-coordinate of the intersection:

y = x + 2

y = -2x + 11

2

What is the minimum value of

y = (x - 4)² - 9

3

How many solutions does the system have?

y = x²

y = x² + 4

4

If

h(x) = 5x + 2

what is h(8)?

5

For what value of k does

y = (x - k)²

have its vertex at x = 6?

Answer Key

1

3

Intersection:

(3, 5)

2

• 9

Vertex:

(4, -9)

3

0

The graphs never intersect.

4

42

Type:

h(8)

5

6

The vertex form already tells you:

y = (x - 6)²