ACT math formulas

to calculate a percentage increase in a product price from a to b

b-a/100 × 100 = c %

standard form of linear equation (slope intercept form)

y = mx + b

what does the y, m, x, and b stand for?

y= dependent variable (output), m= slope, x= independent variable (input), and b= y-intercept (crossing point)

the graph of a quadratic function is a parabola. if the parabola opens upward it is __ ; if it opens downward it is __

positive; negative

in order to find the vertex, you have to find the AOS first. what is the AOS formula?

-b/2a

what does the AOS give you?

the x of the vertex (x,y)

in order to find y in the vertex, you would have to

plug in the x value into the equation

what is an integer?

whole, positive, and negative numbers + 0

even + or - even =

even + or - odd =

even x any integer =

odd x odd =

even x even =

even x odd =

even

odd

even

odd

even

odd

area of a triangle

½ b x h

what does the b and h stand for? what units would the answer be?

b= base length; h= perpendicular height from the base. #cm²

For right angled triangles, use the

Pythagorean theorem (a² + b² = c²)

the sum of interior angles in any triangle is

180

Name the 1st, 2nd, 3rd, and 4th and describe them

diameter= twice the radius

Radius= distance from the center to any point of circumference

Circumference= total distance around the circle

Area of a circle

pie r²

circumference of a circle

2 pie r or pie d

area of a rectangle

l x w

area of a square

s² (the s is one side of the square cuz they all the same)

Volume of a cube

s³

Volume of a rectangular prism

l x w x h

Volume of a cylinder

pie r² h (r is the radius of the base)

what are the units for the volume of a cylinder

#cm³

How do you get the mean

add all the values then divide by the number of values

how do you get the median

Arrange numbers from lowest to highest

• for ODD numbers, the median is the middle

• For EVEN numbers, add the 2 closest to the middle and then divide by 2

⭐️ the median is less affected by the extreme values

what is mode

The most frequent number in the data set

What is straight probability

number of favorable outcome / total number of possibilities

What is the formula for either/or probability

(Outcome A/total number of outcomes) + (outcome B/total number of outcomes)

⭐️called “non-overlapping” probability where it is impossible for 2 or more events to both happen at the same time.

⭐️sum will be the probability of either event happening

• if the 2 (or more) events have the same total numerical of outcomes, just add them both over the total number of outcomes

what is combined probability

(outcome A/total number of outcomes) x (outcome B/total number of outcomes)

⭐️combined probability questions will ultimately have a lower probability then just one (or either) event occurring

• questions will most likely look like, “what are the odds of 2 or more events both/all happening?”

What are simple probability questions like?

they are word problems where you are told a story and asked to find the probability of one or more events

• could be straight, either/or, or a combined probability question

Quadratic formula

-b + or - the square root of (b)² - 4ac all over 2a

• can only solve quadratic equations

the quadratic formula is most helpful when:

factoring doesn’t work

To find the number of solutions for an equation without solving it, use:

the discriminant (b² - 4ac)

• the discriminant tells you whether your solutions are real + distinct, real + equal, or complex

if the discriminant is >, < and = 0, the solutions are

2 real and distinct solutions (crosses the x-axis twice)

• positive values

2 complex (imaginary) solutions (never crosses the x-axis)

• negative values

1 real and equal (repeated) solutions (touches the x-axis once)

• zero values

the slope of a line is

y2 - y1 over x2 - x1

• tells you how steep a line is

⭐️positive slopes: go upward from L to R

⭐️negative slopes: go downward

point-slope form

y-y1 = m(x-x1)

• helpful for writing or identifying line equations quickly without converting to slope intercept form

Distance formula

The square root of (x2-x1)² + (y2-y1)²

• useful for problems that ask you to find the length of a segment or the side of a triangle in a coordinate plane

perimeter of a rectangle

2(l+w)

• this formula and the area of a rectangle, area and circumference of a circle, and the area of a triangle are all crucial for solving problems involving circles, particularly in plane geometry

Pythagorean theorem

a² + b² = c²

• only applies to right triangles where c is the hypotenuse

Midpoint formula

(x1 + x2 over 2 , y1 + y2 over 2)

• useful for finding the center point between 2 coordinates. This can show up in symmetry or shape-division problems.

Here is an example of a function notation question

if f(x) = 2x + 3, then find f(4)

• plug in 4 as x

the equation for a line and parabola is?

• y=mx+b

• y=ax² + bx + c

to find where a line and parabola intersect:

1. Set equations = to each other

2. Rearrange into a quadratic equation

3. Solve for x, whether that is factoring or quadratic formula

4. Plug x into the equation to find y

prime number

a whole number greater than 1 that has only 2 factors: 1 and itself

composite number

a whole number that is greater than 1 and has more than 2 factors

• 2 is the smallest prime number

• 2 is the only even prime number

• 1 is not a prime number

Ex. 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

Even numbers

integers that are exactly divisible by 2

• zero is an even number since 0/2=0

• There are also negative even numbers

Ex. -4, -2, 0, 2, and 4

odd numbers

integers that are not divisible by 2

• There are negative odd numbers

Ex. -5, -3, -1, 1, 3, and 5

Parallel lines have the __ slopes ; perpendicular lines have a __ __ slope

Same ; negative reciprocal

expanded notation form is

writing out what each digit in the number represents

Ex. (5×100) + (9×10) + (3×1) = 593

the 1st arithmetic series sum formula is

Sn = n/2 (a1 + an)

• Sn - the total number of terms added together

• n - number of terms

• a1 - 1st term of sequence

• an - last term of sequence

• d - common difference between the terms

⭐️ use this formula when you know the 1st and last term

The common difference formula is

a(n) - a(n-1)

Fundamental counting principle

method to determine the total number of possible outcomes, w/o having to list them out .

• multiply the # of choices for each event together

right triangles have _____ angle

one 90 degree

equilateral triangles

every side and angle is congruent

• each side is 60 degrees

isosceles triangles

have 2 equal sides and 2 equal angles

scalene triangle

have no congruent sides; each side has a different length

acute triangle

has 3 acute angles

• acute = less than 90 degrees

obtuse triangle

has an obtuse angle

• obtuse = more than 90 degrees

reflection across y-axis flips the __ coordinate

x

Ex. (-2,3) —>

reflection across x-axis flips the __ coordinate

y

Ex. (6,7) —> (6,-7)

what is the perimeter of a rectangle

2L + 2W

what’s the area of a rhombus

what’s the perimeter

½ x diagonal 1 x diagonal 2

4s

what’s the area and perimeter of a kite

• ½ x diagonal 1 x diagonal 2

• add all the sides

what’s the area of a parallelogram

• bh

consecutive angles add up to

180

complementary angles are

90

supplementary angles are

180

what is the area of a trapezoid

½ (base 1 + base 2) h

isosceles triangles have

2 congruent sides and the (bottom) base angles are congruent

for parallelograms, opposite sides are ___ and ___

opposite angles are ___

parallel and congruent

congruent

when diagonals bisect each other, what are they doing

splitting the angles in half

the diagonals of a rectangle are

congruent

for trapezoids, the bases are ___

upper base angles are ___

lower base angles are ___

diagonals are ___

and all 4 angles must = __

parallel

congruent

congruent

congruent

360

• also, the slants on the side are congruent

what is the area and perimeter of a rhombus

½ x diagonal 1 x diagonal 2

4 s

to calculate the height of a triangle, use heron’s formula

the formula is

s = a + b + c over 2

to calculate the area in order to find the height of a triangle (since the area of a triangle is A= 1/2 b h) the formula is

the square root of s(s-a)(s-b)(s-c)

the volume of a pyramid

1/3 x B x H

to find the slant height of a pyramid, use the formula

4 (1/2 (b) (L)

• there’s 4 sides in a pyramid, so that’s why you multiply by 4

the surface area of a pyramid is

(b)² + 4(1/2 (b) (L))

for an equilateral triangle, the surface area is

square root of 3 over 4 x b² + 3(1/2 (b) (L))

if you have the height of the base (in an equilateral triangle), then use this for the area of the base

B = ½ bh

SA = (1/2 b h) + 3 (1/2 b L)

the volume of a cylinder is

pie r² h

the volume of a cone

1/3 pie r² h

to find the arc length in a circle, use this formula

angle/360 = arc length/2 pie r

to find the sector area, use this formula

angle/360 = sector area/pie r²

for degrees of a polygon, the sum of int. angles is

(n-2)180

for the degrees of a polygon, the measure of each angle is

(n-2)180 / n

for the diagonals of a polygon, the number of diagonals is

(n-3)n/2

• use this instead of counting the diagonals

the form of ellipses is

(x-h)²/a² + (y-k)²/b²

• if the ellipse is more horizontal, that means that the a² is bigger

• if the ellipse is more vertical, that means that the b² is bigger

• you can find out the radius of the ellipse by square rooting a or b

hyperbola form is

(x-h)²/a² - (y-k)²/b²

• a = length of hyperbola if it is more horizontal

• b = length of hyperbola if it is more vertical

even functions are reflected over the __ axis

you can reflect it and get __________

f(x)= ____

• x

• the same thing

• f(x)= f(-x)

odd functions are reflected _______

the f(x) function looks like

• at the origin

• -f(x) = f(-x)

y= x² + 1 means

y= x² - 1 means

y= (x+1)² means

y= (x-1)² means

f(-x) means

-f(x) means

• up one (vertical)

• down one (vertical)

• left one (horizontal)

• right one (horizontal)

• over y-axis (reflection)

• over x-axis (reflection)