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Math (SAT®)118Updated Sep 28, 20264 pages

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

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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.
Key Topics: Understanding Slope, Functions, Synthetic Division, etc. – page 1

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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 = 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/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 = 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/B).

Key Topics: Understanding Slope, Functions, Synthetic Division, etc. – page 2

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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(gxx) means we plug the entire gxx 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.

Key Topics: Understanding Slope, Functions, Synthetic Division, etc. – page 3

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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!

Key Topics: Understanding Slope, Functions, Synthetic Division, etc. – page 4

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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/360θ/360 and sector area using πr² · θ/360θ/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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