These study notes cover essential algebra concepts that form the...
Math (SAT®)110 views·Updated Jul 25, 2026·4 pagesMastering SAT Math: Slope, Functions, and Synthetic Division Explained
1 / 4
1
of 4
Linear Equations
Linear equations are the backbone of algebra and appear in several different formats. The most common form is slope-intercept form , where m represents the slope and b is the y-intercept.
You can also write linear equations in standard form (Ax + By = C) or point-slope form . The slope of a line is calculated as the ratio of rise over run (Δy/Δx) or /.
To create an equation from two points, first find the slope, then substitute into slope-intercept form. Next, use one of your points to solve for b, and finally write your complete equation. Remember that horizontal lines have a slope of 0, while vertical lines have an undefined slope.
💡 When converting from standard form (Ax + By = C) to slope-intercept form, solve for y to get: y = -A/B·x + C/B. This immediately shows you the slope and y-intercept (C/B).
2
of 4
Lines and Functions
Functions assign exactly one y-value to each x-value. We write f(3) to represent the y-value when x equals 3, and f(g) means we plug the entire g expression in as the x-value for function f.
To determine if a point (x,y) lies on a line, simply substitute the x and y values into the equation. If the equation becomes a true statement, the point is on the line. For example, checking if (3,4) is on y = 2x: 4 = 2(3) becomes 4 = 6, which is false.
Perpendicular lines have slopes that are negative reciprocals of each other. For instance, if one line has a slope of 2, any perpendicular line will have a slope of -1/2.
🔍 Synthetic division is a shortcut method for dividing polynomials. It's especially useful when applying the remainder theorem—if you divide a polynomial by , the remainder equals the value of the polynomial when x = a.
3
of 4
Quadratics
Quadratic equations (containing x²) can be solved by factoring or using the quadratic formula. The discriminant tells you the nature of the solutions: positive discriminant means two real solutions, zero means one real solution, and negative means complex solutions.
The vertex of a parabola represents its minimum or maximum point. Find the x-coordinate using x = -b/2a or the average of the two roots /2. Then plug this value back into the original equation to find the y-coordinate.
Quadratics can be written in vertex form y = a² + k, where (h,k) is the vertex, or in factored form which clearly shows the x-intercepts. You can also find the sum of roots using -b/a and the product of roots using c/a.
🧩 The vertex form is perfect for finding minimum/maximum values, while the factored form makes it easy to identify where the parabola crosses the x-axis. Choose the form that best suits what you're trying to find!
4
of 4
Complex Numbers and Circles
Complex numbers involve the imaginary unit i, where i² = -1. When working with powers of i, you'll notice a pattern: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats every four powers.
Circle equations follow the form ² + ² = r², where (h,k) is the center point and r is the radius. This is derived from the distance formula between any point on the circle and its center.
For parts of circles, calculate arc length using 2πr · and sector area using πr² · , where θ is the central angle in degrees. These formulas help you work with portions of circles in various applications.
🔄 Complex numbers may seem strange at first, but they become more intuitive with practice. Just remember to treat i like a variable, and follow the pattern of powers as you work through calculations.
We thought you’d never ask...
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar Content
Algebra 2Most popular content: Slope-intercept Form
1Most popular content in Math (SAT®)
9Math (SAT®)Triangle Congruence and Similarity Proofs
Master the application of theorems to prove triangle congruence and similarity in geometric problem-solving.
9th34513
Math (SAT®)Circle Properties: Angles, Chords, and Tangents
Review key theorems and formulas for solving problems involving relationships between angles, chords, and tangent lines within circles for SAT Geometry.
9th3132
Math (SAT®)Calculating Total Surface Area: Prisms, Cylinders, Spheres
Master surface area calculations for prisms, cylinders, and spheres. Includes essential formulas for finding the total surface area of these 3D figures.
9th2497
Math (SAT®)Trigonometric ratios and functions
Trigonometry - Additional Topics in Math - SAT Math
9th1824
Math (SAT®)Calculating percentages
Percentage and Percent Change - Problem Solving and Data Analysis - SAT Math
9th2051
Algebra 1Triangle similarity
Triangle similarity example notes
9th1712
Math (SAT®)Synthetic Division of Polynomials
Learn how to use synthetic and long division to divide polynomials and determine if a divisor is a factor. Understand the remainder theorem and factor theorem.
9th431
Algebra 1ASA Congruent proofs
ASA Congruent proofs notes
9th390
Algebra 1Interior and exterior angle measures
Interior and exterior angle measures practice
9th2721
Most popular content
9I
SAT®Introduction to SAT Error Pattern Analysis
Practice identifying common reasoning traps and misinterpretations in SAT reading and math stimuli to understand why distractors are plausible.
9th2,2050
C
BiologyCell Organelles
This Quiz Is To Test Your Knowledge Of Cell Organelles And Their Functions Inside The Cell. It Can Also Be A Study Guide To Remember Them Better.
6th6950
F
AP PsychologyFoundations of Ethical Guidelines in Research
Practice the core principles of the APA ethical code including informed consent, debriefing, and the role of Institutional Review Boards.
9th1,3370
B
Biologybiology cell organelles and functions
Do you know the cell organelles and their functions?
9th4800
I
SAT®Introduction to SAT Scoring and Scaled Results
Practice interpreting how raw scores are converted to the 1600-point scale and identifying the composition of section scores.
9th8610
F
AP PsychologyFoundations of Research Design and Methodology
Practice distinguishing between different research methods including experiments, correlations, and case studies while identifying key variables.
9th6670
M
MathematicsMath Made Easy: Essential Concepts for Grade 7
Master key math concepts with this comprehensive flashcard set designed specifically for 7th graders. Boost your understanding and ace your exams!
7th7561
M
BiologyMitosis and Cell Division Flashcards
These flashcards cover the basics of mitosis and why cell division occurs in the first place.
9th3470
H
AP PsychologyHistorical Foundations of Psychology
Practice distinguishing between structuralism, functionalism, and the early philosophical roots of psychological science.
9th1,0940
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user
Math (SAT®)110 views·Updated Jul 25, 2026·4 pagesMastering SAT Math: Slope, Functions, and Synthetic Division Explained
These study notes cover essential algebra concepts that form the foundation of high school math. From linear equations and functions to quadratics and complex numbers, mastering these topics will help you solve mathematical problems across various contexts.
1
of 4
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Linear Equations
Linear equations are the backbone of algebra and appear in several different formats. The most common form is slope-intercept form , where m represents the slope and b is the y-intercept.
You can also write linear equations in standard form (Ax + By = C) or point-slope form . The slope of a line is calculated as the ratio of rise over run (Δy/Δx) or /.
To create an equation from two points, first find the slope, then substitute into slope-intercept form. Next, use one of your points to solve for b, and finally write your complete equation. Remember that horizontal lines have a slope of 0, while vertical lines have an undefined slope.
💡 When converting from standard form (Ax + By = C) to slope-intercept form, solve for y to get: y = -A/B·x + C/B. This immediately shows you the slope and y-intercept (C/B).
2
of 4Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Lines and Functions
Functions assign exactly one y-value to each x-value. We write f(3) to represent the y-value when x equals 3, and f(g) means we plug the entire g expression in as the x-value for function f.
To determine if a point (x,y) lies on a line, simply substitute the x and y values into the equation. If the equation becomes a true statement, the point is on the line. For example, checking if (3,4) is on y = 2x: 4 = 2(3) becomes 4 = 6, which is false.
Perpendicular lines have slopes that are negative reciprocals of each other. For instance, if one line has a slope of 2, any perpendicular line will have a slope of -1/2.
🔍 Synthetic division is a shortcut method for dividing polynomials. It's especially useful when applying the remainder theorem—if you divide a polynomial by , the remainder equals the value of the polynomial when x = a.
3
of 4Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Quadratics
Quadratic equations (containing x²) can be solved by factoring or using the quadratic formula. The discriminant tells you the nature of the solutions: positive discriminant means two real solutions, zero means one real solution, and negative means complex solutions.
The vertex of a parabola represents its minimum or maximum point. Find the x-coordinate using x = -b/2a or the average of the two roots /2. Then plug this value back into the original equation to find the y-coordinate.
Quadratics can be written in vertex form y = a² + k, where (h,k) is the vertex, or in factored form which clearly shows the x-intercepts. You can also find the sum of roots using -b/a and the product of roots using c/a.
🧩 The vertex form is perfect for finding minimum/maximum values, while the factored form makes it easy to identify where the parabola crosses the x-axis. Choose the form that best suits what you're trying to find!
4
of 4Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Complex Numbers and Circles
Complex numbers involve the imaginary unit i, where i² = -1. When working with powers of i, you'll notice a pattern: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats every four powers.
Circle equations follow the form ² + ² = r², where (h,k) is the center point and r is the radius. This is derived from the distance formula between any point on the circle and its center.
For parts of circles, calculate arc length using 2πr · and sector area using πr² · , where θ is the central angle in degrees. These formulas help you work with portions of circles in various applications.
🔄 Complex numbers may seem strange at first, but they become more intuitive with practice. Just remember to treat i like a variable, and follow the pattern of powers as you work through calculations.
We thought you’d never ask...
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar Content
Most popular content: Slope-intercept Form
1Most popular content in Math (SAT®)
9Math (SAT®)Triangle Congruence and Similarity Proofs
Master the application of theorems to prove triangle congruence and similarity in geometric problem-solving.
9th34513
Math (SAT®)Circle Properties: Angles, Chords, and Tangents
Review key theorems and formulas for solving problems involving relationships between angles, chords, and tangent lines within circles for SAT Geometry.
9th3132
Math (SAT®)Calculating Total Surface Area: Prisms, Cylinders, Spheres
Master surface area calculations for prisms, cylinders, and spheres. Includes essential formulas for finding the total surface area of these 3D figures.
9th2497
Math (SAT®)Trigonometric ratios and functions
Trigonometry - Additional Topics in Math - SAT Math
9th1824
Math (SAT®)Calculating percentages
Percentage and Percent Change - Problem Solving and Data Analysis - SAT Math
9th2051
Algebra 1Triangle similarity
Triangle similarity example notes
9th1712
Math (SAT®)Synthetic Division of Polynomials
Learn how to use synthetic and long division to divide polynomials and determine if a divisor is a factor. Understand the remainder theorem and factor theorem.
9th431
Algebra 1ASA Congruent proofs
ASA Congruent proofs notes
9th390
Algebra 1Interior and exterior angle measures
Interior and exterior angle measures practice
9th2721
Most popular content
9I
SAT®Introduction to SAT Error Pattern Analysis
Practice identifying common reasoning traps and misinterpretations in SAT reading and math stimuli to understand why distractors are plausible.
9th2,2050
C
BiologyCell Organelles
This Quiz Is To Test Your Knowledge Of Cell Organelles And Their Functions Inside The Cell. It Can Also Be A Study Guide To Remember Them Better.
6th6950
F
AP PsychologyFoundations of Ethical Guidelines in Research
Practice the core principles of the APA ethical code including informed consent, debriefing, and the role of Institutional Review Boards.
9th1,3370
B
Biologybiology cell organelles and functions
Do you know the cell organelles and their functions?
9th4800
I
SAT®Introduction to SAT Scoring and Scaled Results
Practice interpreting how raw scores are converted to the 1600-point scale and identifying the composition of section scores.
9th8610
F
AP PsychologyFoundations of Research Design and Methodology
Practice distinguishing between different research methods including experiments, correlations, and case studies while identifying key variables.
9th6670
M
MathematicsMath Made Easy: Essential Concepts for Grade 7
Master key math concepts with this comprehensive flashcard set designed specifically for 7th graders. Boost your understanding and ace your exams!
7th7561
M
BiologyMitosis and Cell Division Flashcards
These flashcards cover the basics of mitosis and why cell division occurs in the first place.
9th3470
H
AP PsychologyHistorical Foundations of Psychology
Practice distinguishing between structuralism, functionalism, and the early philosophical roots of psychological science.
9th1,0940
Students love us — and so will you.
4.6/5App Store
4.7/5Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
AnnaiOS user