Formula Library

Grade 8 equations explained in plain English.

Students can review slope, linear models, scientific notation, roots, Pythagorean theorem, volume, proportional relationships, and function notation.

Overview Units Equations Practice Parents

Slope Formula

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

Meaning

Slope measures change in y for each 1-unit change in x.

Example

Through (1,4) and (3,10), m=(10-4)/(3-1)=3.

Visual idea

Draw rise and run arrows between points.

Practice

Find slope through (2,5) and (6,13).

Common mistake: Putting change in x over change in y.

Slope-Intercept Form

y=mx+b

Meaning

m is slope and b is the y-intercept.

Example

y=2x+5 starts at 5 and rises 2 for every 1 right.

Visual idea

Start on the y-axis, then move rise/run.

Practice

Graph y=-3x+4.

Common mistake: Plotting b on the x-axis.

Pythagorean Theorem

a^2+b^2=c^2

Meaning

In a right triangle, legs squared add to the hypotenuse squared.

Example

3^2+4^2=25, so c=5.

Visual idea

Build squares on each side of a right triangle.

Practice

Find c if a=8 and b=15.

Common mistake: Using the hypotenuse as a leg.

Scientific Notation

a x 10^n, 1 <= a < 10

Meaning

A compact way to write very large or small numbers.

Example

52,000=5.2 x 10^4.

Visual idea

Use a decimal-place slider.

Practice

Write 0.00043 in scientific notation.

Common mistake: Using 52 instead of 5.2.

Square and Cube Roots

sqrt(64)=8, cuberoot(27)=3

Meaning

Roots undo powers.

Example

x^3=125 has solution x=5.

Visual idea

Use square tiles and cube blocks.

Practice

Solve x^2=49 and x^3=64.

Common mistake: Forgetting x^2=p can have positive and negative solutions.

Cylinder Volume

V=pi r^2h

Meaning

Base area times height.

Example

r=2, h=5 gives V=20pi.

Visual idea

Stack circular layers.

Practice

Find volume when r=3 and h=8.

Common mistake: Using diameter instead of radius.

Cone and Sphere Volume

Vcone=(1/3)pi r^2h, Vsphere=(4/3)pi r^3

Meaning

Cones are one-third of matching cylinders; spheres use radius cubed.

Example

Cone r=3, h=6 has V=18pi.

Visual idea

Compare a cone filling one-third of a cylinder.

Practice

Find sphere volume when r=3.

Common mistake: Forgetting one-third or four-thirds.

Linear Equation Pattern

simplify -> move variables -> move constants -> divide -> check

Meaning

A reliable sequence for one-variable equations.

Example

5x+4=2x+19 becomes 3x=15, so x=5.

Visual idea

Use a balance scale.

Practice

Solve 4x-7=x+11.

Common mistake: Changing only one side.

Proportional Relationship

y=kx

Meaning

A constant unit rate with a graph through the origin.

Example

If tickets cost $8 each, y=8x.

Visual idea

Line through (0,0) with equal steps.

Practice

Write an equation for $12 per hour.

Common mistake: Including a start fee and still calling it proportional.

Function Notation

f(x)=2x+1

Meaning

f(x) names the output when the input is x.

Example

f(3)=2(3)+1=7.

Visual idea

Input-output machine.

Practice

Find f(5) for f(x)=3x-2.

Common mistake: Thinking f(x) means f times x.