Graphing Lab Worksheet Graphing Lab
SAT Math Mastery with Desmos
Student:
Date:
Instructions: Use the Desmos Graphing Calculator to solve the following SAT-style questions. Focus on using graphing to find intersections, analyzing function transformations, and performing complex statistical or algebraic computations. Show any key coordinates or equations you used in the "Work Area".
1
The graphs of the equations \( y = x^2 - 5x + 4 \) and \( y = 2x - 6 \) intersect at two points in the xy-plane. What is the product of the x-coordinates of these two intersection points?
Desmos Work Area: Graph both equations
Product of x-coordinates:
2
The function \( f \) is defined by \( f(x) = -2(x - 3)^2 + 8 \). Which of the following is an x-intercept of the graph of \( y = f(x) \) in the xy-plane?
A) 1
B) 3
C) 5
D) 8
Desmos Strategy: Root finding
3
A list of data values consists of the following numbers: \( \{12, 15, 15, 17, 22, 28, 30, 45, 102\} \). What is the difference between the mean and the median of this data set? (Round to the nearest tenth).
Work Area: Use Desmos list L = [...]
Result:
4
Which of the following points \( (x, y) \) is a solution to the system of inequalities shown below? y > 2x + 1
A) (0, 0)
B) (1, 4)
C) (3, 2)
D) (-2, 6)
Desmos Hint: Graph and check point labels
5
Let \( g(x) = x^3 - 4x \). If the graph of \( h \) is the result of shifting the graph of \( g \) right by 3 units and up by 2 units, what is the value of \( h(2) \)?
Work Area: Write h(x) in terms of g(x) in Desmos
h(2) =
6
A gallery records the following number of visitors over 7 days: \( \{12, 18, 24, 30, 36, 42, 48\} \). If a busy day with 100 visitors is added to the data, what is the positive difference between the median of the original data and the median of the new data?
Work Area: Use median(L) and median(M) in Desmos
Difference:
7
Consider the function \( p(x) = \frac{x^2 - 16}{x - 4} \). What is the y-coordinate of the hole in the graph of \( p \)?
Desmos Method: Zoom in or use tables
y-coordinate:
8
How many distinct solutions are there for the equation \( |2x - 5| = x + 1 \)?
Graph both sides separately in Desmos
Number of Solutions:
9
A circle in the xy-plane has the equation \( x^2 + y^2 - 6x + 8y = 0 \). What is the radius of this circle?
Graph the circle and find the center/edge
Radius:
10
The graphs of \( y = x^3 - 3x^2 + 2 \) and \( y = x - 1 \) intersect at three points. What is the sum of the x-coordinates of these three intersection points?
Show Intersection points here
Sum:
Graphing Lab Answer Key Answer Key
Graphing Lab: SAT Mastery
Teacher Reference Desmos Strategies
Question 1: System Intersections
Ans: 10
Strategy: Graph both equations: \( y = x^2 - 5x + 4 \) and \( y = 2x - 6 \). Click the intersection points. The x-coordinates are 2 and 5. Product = \( 2 \times 5 = 10 \).
Question 2: X-Intercepts
Ans: C (5)
Strategy: Graph \( y = -2(x-3)^2 + 8 \). Click the grey dots where the curve crosses the x-axis. Intercepts are at (1, 0) and (5, 0).
Question 3: Stats Differences
Ans: 11.2
Strategy: Define list \( L = [12, 15, 15, 17, 22, 28, 30, 45, 102] \). Calculate \( mean(L) = 31.77... \) and \( median(L) = 22 \). Difference \( \approx 11.2 \).
Question 4: Inequality Regions
Ans: B (1, 4)
Strategy: Type inequalities into lines 1 and 2. Graph the points A-D as \( (0,0), (1,4), ... \). Point (1,4) lies in the overlapping shaded region.
Question 5: Function Shifts
Ans: -1
Strategy: Line 1: \( g(x) = x^3 - 4x \). Line 2: \( h(x) = g(x-3) + 2 \). Line 3: \( h(2) \). Desmos outputs -1.
Question 6: Median Shift
Ans: 3
Strategy: Define \( L = [12, 18, 24, 30, 36, 42, 48] \). Define \( M = [12, 18, 24, 30, 36, 42, 48, 100] \). Calculate \( median(L) = 30 \) and \( median(M) = 33 \). Difference is 3.
Question 7: Rational Holes
Ans: 8
Strategy: Graph the function. Click and drag along the line to \( x=4 \) to see "undefined". To find the y-value, look at the limit or use a table very close to 4 (e.g., 3.999), which gives 7.999.
Question 8: Absolute Value
Ans: 2
Strategy: Graph \( y = |2x-5| \) and \( y = x+1 \). Count the intersection points. They intersect at \( x = 1.333 \) and \( x = 6 \).
Question 9: Circle Radius
Ans: 5
Strategy: Type the circle equation. Identify the center (3, -4) and a point on the edge like (0,0) or (6,0). Use the distance or just visually count grid units if possible. \( r = \sqrt{3^2 + 4^2} = 5 \).
Question 10: Polynomial Sum
Ans: 3
Strategy: Graph both equations. Click intersections: (-1, -2), (1, 0), and (3, 2). Sum of x-coordinates: \( -1 + 1 + 3 = 3 \).
Desmos Pro Slides SAT Strategy Series
Desmos Power Up
Mastering the Digital SAT Graphing Calculator for High-Speed Problem Solving
The Game Changer
Why Desmos?
On the Digital SAT, the graphing calculator is built right into the app. It's not just a tool—it's a shortcut to correct answers.
Visualizes complex functions instantly
Solves systems without manual algebra
Eliminates computation errors
Insert Desmos UI Screenshot
1 The Intersection Trick
Type ANY system of equations. Desmos finds the solution for you instantly.
Pro Tip
You don't need to isolate \( y \). Just type the equation exactly as it appears in the question.
\( 3x + 4y = 12 \)
\( y = 2x - 5 \)
Click the gray dot to see (x, y)
2 Nonlinear Mastery
Quadratics and Circles are easier when you can see them.
Vertices (Min/Max)
X & Y Intercepts
Asymptotes
Try this in your head vs. Desmos:
\( y = -3(x-1.5)^2 + 12.25 \)
Desmos highlights the vertex and roots automatically!
3 Data Crunching
List Power
Define a list: L = [4, 7, 2, 9, 10]
Then use functions: mean(L), median(L), stdev(L), total(L)
// Calculating mean of 15 numbers
mean(L)
= 6.4
Regression: \( y_1 \sim mx_1 + b \)
When a question asks for a "Line of Best Fit," let Desmos build it using the tilde (~) operator.
4 Transformation Magic
Nested Functions
If \( f(x) = x^2 \), what is \( g(x) = f(x-3) + 2 \)?
Line 1: f(x) = x^2
Line 2: g(x) = f(x-3) + 2
The Slider Hack
Use variables like \( k \) or \( a \) to see functions move in real-time.
"Seeing is believing—and much faster than calculating!"
GO TIME!
Open your Desmos calculators and grab the Graphing Lab Worksheet.
10 Questions
All Calculator Use
Accuracy + Speed = Success
