24 formulas

Algebra 1 Formula Sheet

These are the formulas an Algebra 1 course uses, grouped by unit: lines and slope, quadratics, exponents, growth and money, sequences, and absolute value.

Each formula shows what its letters stand for. Open any one for a worked example and a built-in calculator, or print the whole sheet to study from.

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Lines and slope

Slope Formula
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Calculates the slope (steepness) of a line between two points.
$x_1$
X-coordinate of first point
$y_1$
Y-coordinate of first point
$x_2$
X-coordinate of second point
$y_2$
Y-coordinate of second point
Worked example and calculator
Slope-Intercept Form
$y = mx + b$
The equation of a line where m is the slope and b is the y-intercept.
$m$
Rise over run, the steepness of the line
$b$
Where the line crosses the y-axis
$x$
The independent variable
$y$
The dependent variable
Worked example and calculator
Point-Slope Form
$y - y_1 = m(x - x_1)$
Equation of a line using a known point (x₁, y₁) and the slope m.
$m$
The steepness of the line
$x_1$
X-coordinate of the known point
$y_1$
Y-coordinate of the known point
Worked example and calculator
Standard Form of a Line
$Ax + By = C$
General form of a linear equation.
$A$
Coefficient of x
$B$
Coefficient of y
$C$
Constant term
Worked example and calculator
Parallel & Perpendicular Slopes
$m_{\parallel} = m, \quad m_{\perp} = -\frac{1}{m}$
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
$m$
The slope of the given line
Worked example and calculator
Direct Variation
$y = k x$
y varies directly with x: as x increases, y increases by a constant factor k.
$k$
The fixed ratio y/x
$x$
The independent variable
Worked example and calculator
Distance = Rate × Time
$d = r t$
Distance traveled equals speed (rate) times time, the basis of motion word problems.
$r$
How fast, e.g. miles per hour
$t$
How long, in matching time units
Worked example and calculator

Quadratics and factoring

Quadratic Formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Finds the roots of any quadratic equation ax² + bx + c = 0.
$a$
Coefficient of x²
$b$
Coefficient of x
$c$
The constant value
Worked example and calculator
Discriminant
$\Delta = b^2 - 4ac$
Determines the nature of roots in a quadratic equation.
$a$
Coefficient of x²
$b$
Coefficient of x
$c$
The constant value
Worked example and calculator
Vertex Formula
$x = \frac{-b}{2a}$
Finds the x-coordinate of the vertex of a parabola ax² + bx + c.
$a$
Coefficient of x²
$b$
Coefficient of x
Worked example and calculator
Difference of Squares
$a^{2} - b^{2} = (a + b)(a - b)$
Factorization pattern for difference of two perfect squares.
$a$
First squared term
$b$
Second squared term
Worked example and calculator
Perfect Square Trinomial
$a^2 \pm 2ab + b^2 = (a \pm b)^2$
Factorization pattern for perfect square trinomials.
$a$
First squared term
$b$
Second squared term
Worked example and calculator

Exponents and scientific notation

Product Rule for Exponents
$a^m \cdot a^n = a^{m+n}$
When multiplying with same base, add the exponents.
$a$
The common base
$m$
First exponent
$n$
Second exponent
Worked example and calculator
Quotient Rule for Exponents
$\frac{a^m}{a^n} = a^{m-n}$
When dividing with same base, subtract the exponents.
$a$
The common base
$m$
Exponent in numerator
$n$
Exponent in denominator
Worked example and calculator
Power Rule for Exponents
$(a^m)^n = a^{mn}$
When raising a power to a power, multiply the exponents.
$a$
The base
$m$
Exponent inside parentheses
$n$
Exponent outside parentheses
Worked example and calculator
Scientific Notation
$N = a \times 10^n \quad (1 \le a < 10)$
Write very large or small magnitudes as a coefficient times a power of ten.
$a$
One-digit (or 1.0–9.9…) factor with 1 ≤ a < 10
$n$
Integer power of 10
Worked example and calculator

Growth, percent and money

Exponential Growth & Decay
$A = a(1 + r)^t$
A quantity that grows (r > 0) or decays (r < 0) by a fixed percent each period. Enter r as a decimal.
$a$
Starting value at t = 0
$r$
Decimal rate; positive for growth, negative for decay
$t$
Number of periods elapsed
Worked example and calculator
Percent Change
$\text{percent change} = \frac{V_2 - V_1}{V_1}\cdot 100\%$
How much a value increased or decreased, as a percent of the original. Positive is an increase, negative a decrease.
$V_1$
The starting (old) amount
$V_2$
The ending (new) amount
Worked example and calculator
Simple Interest
$I = P r t$
Interest earned (or owed) on a principal P at annual rate r for t years. Enter r as a decimal (0.05 for 5%).
$P$
Starting amount of money
$r$
Interest rate as a decimal, e.g. 0.05 for 5%
$t$
Length of time in years (years)
Worked example and calculator
Compound Interest
$A = P\left(1 + \frac{r}{n}\right)^{n t}$
Final balance when interest compounds n times per year. Enter r as a decimal (0.05 for 5%).
$P$
Starting amount of money
$r$
Interest rate as a decimal, e.g. 0.05 for 5%
$n$
Compounds per year (12 = monthly, 4 = quarterly)
$t$
Length of time in years (years)
Worked example and calculator

Sequences

Arithmetic Sequence (nth Term)
$a_n = a_1 + (n - 1)d$
Each term adds a common difference d to the previous one; aₙ is the value at position n.
$a_n$
Term at index n
$a_1$
Starting value
$n$
Term number (positive integer)
$d$
Added each step
Worked example and calculator
Geometric Sequence (nth Term)
$a_n = a_1 r^{n - 1} \quad (r \neq 0 \text{ typical})$
Each term multiplies the previous by common ratio r; growth or decay by a constant factor.
$a_n$
Term at index n
$a_1$
Starting value
$r$
Factor between successive terms
$n$
Term number
Worked example and calculator

Absolute value

Absolute Value Inequality |x| < a
$|x| < a \iff -a < x < a \quad (a > 0)$
For positive a, |x| < a means x is strictly between −a and a, a bounded interval.
$x$
The unknown
$a$
Positive constant bound
Worked example and calculator
Absolute Value Inequality |x| > a
$|x| > a \iff x < {-a} \;\text{ or }\; x > a \quad (a > 0)$
For positive a, |x| > a means x is farther than a from 0, two rays, not a single interval.
$x$
The unknown
$a$
Positive constant bound
Worked example and calculator

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Algebra 1 Formula Sheet

These are the formulas an Algebra 1 course uses, grouped by unit: lines and slope, quadratics, exponents, growth and money, sequences, and absolute value.

Each formula shows what its letters stand for. Open any one for a worked example and a built-in calculator, or print the whole sheet to study from.

Lines and slope

Slope Formula

m = \frac{y_2 - y_1}{x_2 - x_1}

Calculates the slope (steepness) of a line between two points.

$x_1$: X-coordinate of first point
$y_1$: Y-coordinate of first point
$x_2$: X-coordinate of second point
$y_2$: Y-coordinate of second point

Lines and slope

Slope Formula

Slope-Intercept Form

Point-Slope Form

Standard Form of a Line

Parallel & Perpendicular Slopes

Direct Variation

Distance = Rate × Time

Quadratics and factoring

Quadratic Formula

Discriminant

Vertex Formula

Difference of Squares

Perfect Square Trinomial

Exponents and scientific notation

Product Rule for Exponents

Quotient Rule for Exponents

Power Rule for Exponents

Scientific Notation

Growth, percent and money

Exponential Growth & Decay

Percent Change

Simple Interest

Compound Interest

Sequences

Arithmetic Sequence (nth Term)

Geometric Sequence (nth Term)

Absolute value

Absolute Value Inequality |x| < a

Absolute Value Inequality |x| > a

Learn these formulas

Other formula sheets

Lines and slope

Slope Formula

Slope-Intercept Form

Point-Slope Form

Standard Form of a Line

Parallel & Perpendicular Slopes

Direct Variation

Distance = Rate × Time

Quadratics and factoring

Quadratic Formula

Discriminant

Vertex Formula

Difference of Squares

Perfect Square Trinomial

Exponents and scientific notation

Product Rule for Exponents

Quotient Rule for Exponents

Power Rule for Exponents

Scientific Notation

Growth, percent and money

Exponential Growth & Decay

Percent Change

Simple Interest

Compound Interest

Sequences

Arithmetic Sequence (nth Term)

Geometric Sequence (nth Term)

Absolute value

Absolute Value Inequality |x| < a

Absolute Value Inequality |x| > a

Learn these formulas

Other formula sheets