Fiveable SAT Math: Formulas to Memorize

SAT Math: Essential Formulas To Memorize

Looking for a list of formulas to memorize for the SAT Math section? 👀

You're in the right place! We've compiled a list of formulas that are helpful to memorize when tackling the SAT Math section. Let's get into it and break down each formula, grab your notebook! 📒

SAT Math: Linear Line Formulas

First, we're going to dive into some formulas for straight lines. Linear equations can exist in three main forms, and it's important to know when to use which to help you answer a question!

The Standard Form of Linear Equations

🏹 Standard form: Ax + By = C

• • "A": coefficient of x
• • "B": coefficient of y
• • "x" and "y": variables
• • "C": constant

🤔The standard form of a linear equation represents a line as a combination of x and y variables with coefficients (A and B) and a constant (C). It provides a general form for linear equations, but it is often rearranged to other forms for easier interpretation.

The Slope-Intercept Form of Linear Equations

🏹Slope-Intercept form: y= mx+b 

• • "m": slope of the line
• • "y" and "x": variables
• • "b": y-intercept of the line

🤔The slope-intercept form is a commonly used representation of a linear equation. It shows the relationship between the y-coordinate and the x-coordinate on the line. The slope (m) represents the line's steepness, and the y-intercept (b) is the point where the line intersects the y-axis.

The Point-Slope Form of Linear Equations

🏹Point-Slope form: y - y₁ = m(x - x₁) 

• • "x₁" and "y₁" : coordinates of a given point
• • "m": slope
• • "y" and "x": variables

🤔The point-slope form of a linear equation (y - y₁ = m(x - x₁)) is useful for determining the equation of a line when the slope (m) and a point (x₁, y₁) on the line are known.

The Slope of Linear Lines

Remember how slope equals rise/run? Or m=rise/run? Let's take a look at what this really means!

🏹Slope of a linear line: (y₂ - y₁) / (x₂ - x₁)

• • "y₁" and "x₁": the coordinates of the first point on the line
• • "y₂" and "x₂": the coordinates of the second point on the line- - It really doesn't matter which point comes first and which comes second, just make sure that x₁ and y₁ are the coordinates of the same point; and vice versa for x₂ and y₂.

🤔The slope formula calculates the slope of a line between two points (x₁, y₁) and (x₂, y₂) by finding the ratio of the vertical change (rise) to the horizontal change (run). 

The Midpoint Formula

🏹Midpoint formula: ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2) 

• • (x₁ + x₂) / 2 is used to find the x-coordinate while the (y₁ + y₂) / 2 is used to find the y-coordinate of the midpoint. The answer should be written in (x,y) form.
• • "y₁" and "x₁": the coordinates of the first point on the line
• • "y₂" and "x₂": the coordinates of the second point on the line

🤔The midpoint formula helps find the coordinates of the midpoint between two given points. By averaging the x-coordinates and the y-coordinates of the two points, we can determine the coordinates of the midpoint. This formula allows us to determine the center point or middle point on a line segment. 

The Distance Formula

🏹Distance Formula: √((x₂ - x₁)² + (y₂ - y₁)²)

• • "y₁" and "x₁": the coordinates of the first point on the line
• • "y₂" and "x₂": the coordinates of the second point on the line

🤔The distance formula calculates the distance between two points on a coordinate plane. By using the coordinates of the two points, it determines the length of the line segment connecting them. This formula utilizes the Pythagorean theorem to find the square root of the sum of the squares of the differences between the x-coordinates and the y-coordinates.

SAT Math: Distance/Rate Formula

🏹 Distance = Speed × Time

• • "Distance": the total distance traveled by the object
• • "Speed": the rate at which the object is moving
• • "Time": the duration of the travel

🤔The distance/rate formula is a fundamental formula used to calculate the distance traveled by an object based on its speed and the time it takes to travel. By using this formula, we can analyze and solve various problems related to distance, speed, and time. It is commonly applied in physics, everyday travel calculations, and other fields where measuring and understanding distances and rates of motion are important.

SAT Math: Quadratic Equations/Parabolas

Just like the linear line equations, there are several forms of quadratic equations. Let's dig in! 🤓

Standard Form of a Quadratic Equation

🏹Standard form: ax² + bx + c = 0

• • "a", "b", and "c": constants- - "a" determines the shape of the parabolic curve. If "a" is positive, the parabola opens upward, and if "a" is negative, it opens downward.- - "b" and "c" affect the position and orientation of the parabola.
• • "x": variable

🤔The standard/quadratic form of a quadratic equation represents a parabola as a quadratic expression equal to zero. The constants "a," "b," and "c" determine the shape, position, and orientation of the parabola. The coefficient "a" determines whether the parabola opens upward or downward, while "b" and "c" affect its position and orientation. 

The Vertex Form of a Quadratic Equation

🏹 Vertex form: y = a(x - h)² + k

• • "a": coefficient of the quadratic term
• • "(h, k)": the coordinates of the vertex- - h: x-coordinate of the vertex- - k: y-coordinate of the vertex
• • "x": variable

🤔The vertex form of a quadratic equation represents a parabola in terms of its vertex coordinates, (h, k), and the coefficient "a." The vertex (h, k) is the point where the parabola reaches its maximum or minimum value, depending on whether "a" is positive or negative. 

The Factored Form of a Quadratic Equation

🏹 Factored form: y = a(x - r₁)(x - r₂)

• • "a": coefficient of the quadratic term
• • "r₁" and "r₂": the roots or solutions of the quadratic equation- - "r₁" and "r₂" represent the x-coordinates where the graph intersects the x-axis. In simple terms, the roots are the values of "x" that make the equation equal to zero.

🤔The factored form of a quadratic equation represents the equation as a product of linear factors (x - r₁)(x - r₂), where "r₁" and "r₂" are the roots or solutions of the equation. These roots are the x-coordinates where the graph of the quadratic equation intersects the x-axis, meaning they are the values of "x" that make the equation equal to zero. The factored form allows us to easily identify the roots and understand how the quadratic equation is factored.

Coordinates of the Vertex in a Parabola

🏹The x-Coordinate of the Vertex: x = -b / (2a)

🏹The y-Coordinate of the Vertex: y = f( -b / (2a)) 

• • Here, "a" and "b" are the coefficients of the quadratic equation: ax²+by+c. The x-coordinate of the vertex represents the axis of symmetry of the parabola and is obtained by dividing "-b" by 2 times the coefficient "a".
• • To find the y-coordinate of the vertex, plug in the x-coordinate obtained above into the equation as x. The resulting value will give you the y-coordinate of the vertex.

🤔The coordinates of the vertex provide information about the vertex of the parabolic curve represented by the quadratic equation: 

• • The x-coordinate of the vertex, given by x = -b / (2a), represents the axis of symmetry of the parabola. It is obtained by dividing the negative coefficient "b" by 2 times the coefficient "a."
• • The y-coordinate of the vertex, denoted as y = f(-b / (2a)), is obtained by substituting the x-coordinate into the original equation. This provides the corresponding y-value of the vertex.

SAT Math: Circle Formulas

Arc Length Formula: L = 2πr (θ/360)

• • "L": length
• • "r": radius
• • "θ": the measure of the central angle subtended by the arc

Sector Area Formula: A = (θ/360)πr²

• • "A": area
• • "r": radius
• • "θ": measure of the central angle subtended by the sector

Center-Radius Equation: (x - h)² + (y - k)² = r²

• • "(h, k)": coordinates of the center of the circle- - "h": x-coordinate- - "k": y-coordinate
• • "r": the radius—the distance from the center to any point on the circle.
• • "x" and "y": variables

SAT Math: Exponents/ Roots Formulas

Product of Powers: a^m * a^n = a^(m + n)

• • "a": the base
• • "m" and "n": the exponents or powers associated with that base

Power of a Power: (a^m)^n = a^(m * n)

• • "a": the base
• • "m" and "n": the exponents or powers associated with that base

Power of a Product: (a * b)^n = a^n * b^n

• • "a" and "b": the base
• • "n": the exponent associated with the base

Quotient of Powers: a^m / a^n = a^(m - n)

• • "a": the base
• • "m" and "n": the exponents or powers associated with that base

Negative Exponent: a^(-n) = 1 / a^n

• • "a": the base
• • "n": the exponent associated with the base

SAT Math: Exponential Function/ Compound Interest

General form of an exponential function: f(x) = a * b^x

• • "f(x)": represents the output or value of the function at a given input
• • "x": input
• • "a" : the initial value or y-intercept of the function
• • "b": the base of the exponential function- - If the base "b" is greater than 1, the function exhibits exponential growth as "x" increases.- - If the base "b" is between 0 and 1, the function shows exponential decay as "x" increases.- - If the base "b" is equal to 1, the function becomes a constant function with a horizontal line.

Compound interest formula: A = P(1 + r/n)^(nt)

• • "A": total amount
• • "P": the principal (initial) amount
• • "r": the annual interest rate (expressed as a decimal),
• • "n": the number of compounding periods per year
• • "t": the number of year

Continuous compound interest formula: A = P * e^(rt)

• • "A": total amount
• • "P": the principal (initial) amount
• • "e": Euler's number (approximately 2.71828)
• • "r": the annual interest rate (expressed as a decimal)
• • "t": the number of year

SAT Math: Trigonometry Functions

Image Courtesy of MathsisFun.

Sine (sin): The sine of an angle is the length of the side opposite to the angle divided by the length of the hypotenuse. 

• • sin(θ) = opposite/hypotenuse

Cosine (cos): The cosine of an angle is the length of the side adjacent to the angle divided by the length of the hypotenuse.

• • cos(θ) = adjacent/hypotenuse

Tangent (tan): The tangent of an angle is the length of the side opposite the angle divided by the length of the side adjacent to the angle.

• • tan(θ) = opposite/adjacent

Cosecant (csc): The cosecant of an angle is the reciprocal of the sine of the angle.

• • csc(θ) = 1/sin(θ)

Secant (sec): The secant of an angle is the reciprocal of the cosine of the angle 

• • sec(θ) = 1/cos(θ)

Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent of the angle.

• • cot(θ) = 1/tan(θ)

And there you have it! We've explored a variety of important math formulas that will not only help you excel in the SAT but also expand your mathematical toolkit. So go forth, my math-savvy friend, and let the formulas guide you toward success! Good luck, and may your SAT experience be filled with joy and achievement.