Looking for a list of formulas to memorize for the SAT Math section? ๐
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! ๐
**It is important to note that while the math reference sheet provided in the SAT is incredibly helpful, it doesn't include all the formulas you'll need to know. So, while the reference sheet provides a solid starting point, our goal is to dive deeper into the world of math formulas and thereby successfully prep you for that rigorous SAT math section. **
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!
๐น Standard form: Ax + By = C
๐คThe 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.
๐นSlope-Intercept form: y= mx+b
๐คThe 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.
๐นPoint-Slope form: y - yโ = m(x - xโ)
๐คThe point-slope form of a linear equation (y - yโ = m(x - xโ)) is useful for determining the equation of a line when the slope (m) and a point (xโ, yโ) on the line are known.
Remember how slope equals rise/run? Or m=rise/run? Let's take a look at what this really means!
๐นSlope of a linear line: (yโ - yโ) / (xโ - xโ)
๐คThe slope formula calculates the slope of a line between two points (xโ, yโ) and (xโ, yโ) by finding the ratio of the vertical change (rise) to the horizontal change (run).
๐นMidpoint formula: ( (xโ + xโ) / 2 , (yโ + yโ) / 2)
๐คThe 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.
๐นDistance Formula: โ((xโ - xโ)ยฒ + (yโ - yโ)ยฒ)
๐คThe 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.
๐น Distance = Speed ร Time
๐คThe 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.
You may be wondering what the difference is between this distance formula and the one we included in the Linear Line formulas! Essentially, the linear line distance formula helps us find the distance between two points on a graph, while this distance formula is used when calculating the distance traveled by an object when it moves at a constant speed.
Just like the linear line equations, there are several forms of quadratic equations. Let's dig in! ๐ค
๐นStandard form: axยฒ + bx + c = 0
๐คThe 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.
๐น Vertex form: y = a(x - h)ยฒ + k
๐คThe 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.
๐น Factored form: y = a(x - rโ)(x - rโ)
๐คThe factored form of a quadratic equation represents the equation as a product of linear factors (x - rโ)(x - rโ), where "rโ" and "rโ" 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.
๐นThe x-Coordinate of the Vertex: x = -b / (2a)
๐นThe y-Coordinate of the Vertex: y = f( -b / (2a))
๐คThe coordinates of the vertex provide information about the vertex of the parabolic curve represented by the quadratic equation:
Arc Length Formula: L = 2ฯr (ฮธ/360)
** The arc length formula calculates the length (L) of an arc on a circle based on the radius (r) and the measure of the central angle (ฮธ) subtended by that arc. It relates the length of the arc to the circumference of the circle.**
Sector Area Formula: A = (ฮธ/360)ฯrยฒ
** The sector area formula calculates the area (A) of a sector of a circle using the radius (r) and the measure of the central angle (ฮธ) subtended by that sector. It relates the area of the sector to the total area of the circle.**
Center-Radius Equation: (x - h)ยฒ + (y - k)ยฒ = rยฒ
** The center-radius equation represents the equation of a circle with its center at the point (h, k) and a radius (r). It shows the relationship between the coordinates (x, y) on the circle and the distance from the center to any point on the circle.**
Product of Powers: a^m * a^n = a^(m + n)
** The product of powers formula states that when you multiply two powers with the same base (a^m * a^n), you can add their exponents to obtain the result (a^(m + n)). It allows for the consolidation of like terms and simplifies the expression. **
Power of a Power: (a^m)^n = a^(m * n)
** The power of a power formula states that when you raise a power to another exponent ((a^m)^n), you can multiply the exponents to obtain the result (a^(m * n)). It demonstrates the relationship between nested exponents and simplifies the expression.**
Power of a Product: (a * b)^n = a^n * b^n
** The power of a product formula states that when you raise a product of bases to an exponent ((a * b)^n), you can distribute the exponent to each base to obtain the result (a^n * b^n). It allows for the individual evaluation of powers on each base and simplifies the expression.**
Quotient of Powers: a^m / a^n = a^(m - n)
** The quotient of powers formula states that when you divide two powers with the same base (a^m / a^n), you can subtract the exponents to obtain the result (a^(m - n)). This formula allows you to simplify expressions involving the division of powers with the same base by subtracting their exponents.**
Negative Exponent: a^(-n) = 1 / a^n
** The negative exponent formula states that when you have a negative exponent (a^(-n)), you can rewrite it as the reciprocal of the base raised to the positive exponent (1 / a^n). This formula allows you to convert a negative exponent into a positive exponent and express the result as a fraction. **
General form of an exponential function: f(x) = a * b^x
** The general form of an exponential function represents a mathematical relationship between an input (x) and its corresponding output or value (f(x)). **
Compound interest formula: A = P(1 + r/n)^(nt)
** Exponential functions find various applications, and one notable example is compound interest. In the context of compound interest, the general form of an exponential function is used to model the growth of an investment over time.**
Continuous compound interest formula: A = P * e^(rt)
** The continuous compound interest formula is used to calculate the total amount (A) accumulated through continuous compounding of interest. By applying the continuous compound interest formula, you can determine the future value of an investment that earns continuous compounding interest over time. Continuous compounding assumes that the interest is constantly reinvested and compounded without any specific intervals or discrete periods.**
Sine (sin): The sine of an angle is the length of the side opposite to the angle divided by the length of the 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.
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.
Cosecant (csc): The cosecant of an angle is the reciprocal of the sine of the angle.
Secant (sec): The secant of an angle is the reciprocal of the cosine of the angle
Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent of the angle.
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.