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Math (SAT®)Math (SAT®)109 views·Updated May 23, 2026·4 pages

Mastering SAT Math: Slope, Functions, and Synthetic Division Explained

user profile picture
kendall :)@kendall_moqo

These study notes cover essential algebra concepts that form the... Show more

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of 4
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

Linear Equations

Linear equations are the backbone of algebra and appear in several different formats. The most common form is slope-intercept form y=mx+by = mx + b, where m represents the slope and b is the y-intercept.

You can also write linear equations in standard form Ax+By=CAx + By = C or point-slope form y−y1=m(x−x1)y - y₁ = m(x - x₁). The slope of a line is calculated as the ratio of rise over run Δy/ΔxΔy/Δx or y2−y1y₂-y₁/x2−x1x₂-x₁.

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=CAx + By = C to slope-intercept form, solve for y to get: y = -A/B·x + C/B. This immediately shows you the slope −A/B-A/B and y-intercept C/BC/B.

2
of 4
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

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(x)) means we plug the entire g(x) 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 x−ax-a, the remainder equals the value of the polynomial when x = a.

3
of 4
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

Quadratics

Quadratic equations (containing x²) can be solved by factoring or using the quadratic formula. The discriminant b2−4acb² - 4ac 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 x1+x2x₁ + x₂/2. Then plug this value back into the original equation to find the y-coordinate.

Quadratics can be written in vertex form y = ax−hx - h² + 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
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

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 x−hx-h² + y−ky-k² = 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 · (θ/360) and sector area using πr² · (θ/360), 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 i,−1,−i,1i, -1, -i, 1 as you work through calculations.

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Math (SAT®)Math (SAT®)109 views·Updated May 23, 2026·4 pages

Mastering SAT Math: Slope, Functions, and Synthetic Division Explained

user profile picture
kendall :)@kendall_moqo

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
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

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 y=mx+by = mx + b, where m represents the slope and b is the y-intercept.

You can also write linear equations in standard form Ax+By=CAx + By = C or point-slope form y−y1=m(x−x1)y - y₁ = m(x - x₁). The slope of a line is calculated as the ratio of rise over run Δy/ΔxΔy/Δx or y2−y1y₂-y₁/x2−x1x₂-x₁.

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=CAx + By = C to slope-intercept form, solve for y to get: y = -A/B·x + C/B. This immediately shows you the slope −A/B-A/B and y-intercept C/BC/B.

2
of 4
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

Sign 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(x)) means we plug the entire g(x) 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 x−ax-a, the remainder equals the value of the polynomial when x = a.

3
of 4
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

Sign 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 b2−4acb² - 4ac 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 x1+x2x₁ + x₂/2. Then plug this value back into the original equation to find the y-coordinate.

Quadratics can be written in vertex form y = ax−hx - h² + 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
# Linear EQUATIONS constant table/X'equation Slope -intercept form Y-mx-b Standard form Ax + By: C Point-slope form 4-4.=m(x-x.) Slope $ \

Sign 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 x−hx-h² + y−ky-k² = 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 · (θ/360) and sector area using πr² · (θ/360), 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 i,−1,−i,1i, -1, -i, 1 as you work through calculations.

We thought you’d never ask...

What is the Knowunity AI companion?

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.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content: Slope-intercept Form

4

Most popular content in Math (SAT®)

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

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