sat formulas

slope intercept form

y = mx + b

use to write, graph, or interpret linear equations where m is the slope and b is the y-intercept (the starting value when x = 0).

slope formula & desmos trick

m = (y2 - y1) / (x2 - x1)

use when given two coordinate points (x1, y1) and (x2, y2) and you need to find the slope

desmos trick: click the "+" button, add a table, and enter your coordinate points into x1 and y1. In the next line, type "y1 ~ mx1 + b" (using the tilde key). Desmos instantly gives you the exact slope (m) and y-intercept (b). you could also click the regression symbol, pick linear regression (ONLY if it’s linear), and it’ll do the same thing

point slope form

y - y1 = m(x - x1)

quickly construct a line's equation when you know one specific point on the line and the slope, without having to calculate b first

quadratic formula

x = (-b ± √(b² - 4ac)) / (2a)

used to find x-intercepts or solutions (roots) of any quadratic equation ax² + bx + c = 0 when it can't be easily factored.

desmos: just put the quad equation into desmos and look to see where the graph hits 0

discriminant

D = b² - 4ac

used when the question asks how many real solutions a quadratic equation has (if D > 0, 2 solutions; if D = 0, 1 solution; if D < 0, 0 real solutions)

desmos: js put the graph into desmos n look.. but the equation is good if you wanna be 100% sure about the real vs imaginary solutions

vertex form of a quadratic

y = a(x - h)² + k

use to immediately identify the highest or lowest point (the vertex (h, k)) or the axis of symmetry (x = h) of a parabola

desmos: put it in the graph…

sum n product of roots shortcut

sum = -b/a

product = c/a

used when asked to find the sum or product of all solutions to ax² + bx + c = 0 directly without solving for x first

exponential growth and decay

growth: y = a(1 + r)^t

decay: y = a(1 - r)^t

word problems involving population growth, interest rates, or depreciation where an initial amount a increases (+) or decreases (-) by a percentage r over time t.

simple interest formula

I = P r t

how to solve system of equations?

desmos: type one equation on one line and the other equation on a different line, and see where the two intersect. also works when a question shows two equations equaling each other

how to find an exponential or quadratic equation from a word problem?

exponential model: y = ab^t (t=time)

quadratic model y = ax^2 + bx + c

desmos: add a table with your data points. to find an exponential equation, type "y1 ~ a(b)^x1". to find a quadrati , type "y1 ~ ax1^2 + bx1 + c"

how do you solve for an equation with an unknown constant?

type the equation into Desmos using "k" instead of a number. Desmos will automatically prompt you to "Add Slider for k". Drag the slider or click the number to adjust k until the lines coincide (overlap completely) to find the exact value of k.

how do you evaluate cofunction identities or trig values without memorizing conversion rules?

concept: sin(x) = cos(90-x)

concept: sin(x)² + cos(x)² = 1

click the wrench icon in the top right of desmos n switch the angle mode from “radians” to “degrees” (make sure you switch it back after so you don’t mess up the calculations for other questions!!!) you can type any trig expression directly (for example "sin(30)") or test identities by typing them to see if they evaluate correctly

circle area formula (desmos cant do)

A = pi * r^2

circle circumference

C = 2*pi*r

equation of a circle

(x - h)^2 + (y - k)^2 = r^2

cylinder volume

V = pi*r^2*h

cone volume

V = (1/3) * pi * r² * h

sphere volume

V = (4/3) * pi * r³

how do you evaluate function values or find composite functions without manual substitution?

algebraic: plug the input value into the function variable and evaluate step-by-step using order of operations.

desmos: type the function into desmos (for example: f(x) = 3x^2 - 5x). on the next line, type f(4) to get the answer

(situations where they give you something like f(a+1) are possible on desmos, but at that point it’s easier to just do the algebra. so it would be 3(a+1)² - 5(a+1) and you’d just have to distribute and solve)

how can you check if two algebraic expressions are equivalent or find which answer choice matches a complex expression?

desmos: type the equation in the question in desmos, and then on a second line type out each answer choice. the answer that perfectly overlaps with the original equation is the correct one

how do you solve absolute value equations or find the number of solutions?

desmos: type both sides of the equation as separate functions (like y = |2x - 4| and y = 6). Click the intersection points to instantly find all solutions without needing to split cases.

how do you determine which point (x, y) satisfies a system of linear inequalities?

type the inequalities into desmos (using <, >, <=, >=). desmos will shade the regions. check each answer choice is found in that double shaded region

how to find sum/product of the zeros of a quadratic?

graph the quadratic in desmos, click the x-intercepts and add/multiply depending on what theyre asking

how do you handle radical equations (square roots with variables)?

type the radical equation as a function or graph both sides (like y = sqrt(x + 5) and y = x - 1). click the intersection point to find the true solution

make sure on desmos you click on the square root button rather than typing sqrt!

how do you find the maximum or minimum value in a word problem (like max profit or max height)?

type the equation into desmos, zoom out if needed, and simply click the highest peak (maximum) or lowest valley (minimum) of the curve to read the coordinates immediately.

how to find an unknown constant (like k) that makes a system have infinitely many solutions or no solution?

1. type the equations into Desmos containing the letter constant, and desmos will prompt you to add a slider for k

2. click and drag the slider, or click the slider number to adjust it until the lines overlap completely (infinitely many solutions) or stay parallel (no solution). for higher or lower values, type it in first to expand the slider

how do you quickly find the roots, zeros, or solutions of a high-degree polynomial equation like x^3 - 4x^2 + x + 6 = 0?

1. move everything to one side so the equation equals zero, then type the non-zero side as y = [expression].

2. look at the graph and click where the curve crosses the x-axis (y = 0). 3. The x-coordinates of those points are your roots/solutions.

how do you find complex roots, maximums, or minimums of equations that do not cross the x-axis or when desmos hides the peak?

1. f an equation has no real solutions (no x-intercepts), desmos will not show any points on the x-axis

2. instead, use the vertex form or look at the table (click the gear icon next to the equation and convert to a table) to find the exact turning point (maximum or minimum y-value).

3. if desmos still doesn’t show it, you can solve it algebraically: x = -(b/2a). and then plug the answer into the og equation to get the y

4. for complex roots, use the quadratic formula

when to not use desmos?

1. when a question asks you to rearrange a formula like "solve for t in terms of r" where no numbers are given, desmos cant solve for abstract variables

2. when problems have no numbers and only letters

3. some probility problems should just be solved by hand

4. if you spend too much time trying to find things using a slider (if there are no answer choices)