Math · Middle school and high school algebra
Linear Equations and Functions Study Guide
A linear equation graphs as a straight line, and its slope-intercept form, y = mx + b, gives the two things you need to draw it: the slope m and the y-intercept b. A function is a rule that gives exactly one output for each input, and every non-vertical line is a function. Most problems ask you to find a slope, write an equation from two points or a graph, or decide whether a relation is a function.
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Play Linear Equations DemoKey terms
- Linear equation
- An equation whose graph is a straight line. In two variables it can be written as y = mx + b.
- Slope
- The steepness of a line: rise over run, or the change in y divided by the change in x.
- Y-intercept
- The point where a line crosses the y-axis, where x equals zero.
- X-intercept
- The point where a line crosses the x-axis, where y equals zero.
- Slope-intercept form
- The equation y = mx + b, where m is the slope and b is the y-intercept.
- Point-slope form
- The equation y − y₁ = m(x − x₁), used when you know the slope and one point on the line.
- Standard form
- The equation Ax + By = C, where A, B and C are numbers.
- Function
- A relation in which each input has exactly one output.
- Domain
- The set of all possible input (x) values of a function.
- Range
- The set of all possible output (y) values of a function.
- Independent variable
- The input of a function, usually x. Its value is chosen freely.
- Dependent variable
- The output of a function, usually y. Its value depends on the input.
- Parallel lines
- Lines in the same plane that never meet. They have the same slope.
- Perpendicular lines
- Lines that meet at a right angle. Their slopes are negative reciprocals of each other.
Finding the slope and writing an equation
The slope between two points is m = (y₂ − y₁) ÷ (x₂ − x₁). A positive slope rises from left to right, a negative slope falls, a horizontal line has slope zero, and a vertical line has an undefined slope.
- 1Find the slope from the two points.
- 2Substitute the slope and one point into y = mx + b and solve for b.
- 3Write the equation with the m and b you found.
- 4Check it by substituting the other point.
Three forms of the same line
- •Slope-intercept form, y = mx + b: best for graphing from the slope and y-intercept.
- •Point-slope form, y − y₁ = m(x − x₁): best when you know a point and the slope.
- •Standard form, Ax + By = C: best for finding intercepts.
What makes a function
A relation is a function if no input has two different outputs. On a graph, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function. Function notation, such as f(x) = 2x + 3, names the rule, so f(4) = 11.
Reading a word problem
In a linear model the slope is the rate of change and the y-intercept is the starting value. A plan that costs $5 per hour plus a $20 fee is y = 5x + 20.
Slope = how fast it changes. Intercept = where it starts.
Common mistakes
Mistake: Slope is run over rise.
Slope is rise over run: the change in y divided by the change in x.
Mistake: A vertical line has a slope of zero.
A horizontal line has slope zero. A vertical line has an undefined slope.
Mistake: Using the points in a different order on top and bottom of the slope formula.
Subtract in the same order in both: (y₂ − y₁) over (x₂ − x₁). Flipping one of them flips the sign.
Mistake: A relation is not a function if two inputs have the same output.
That is allowed. A relation fails to be a function only when one input has two different outputs.
Mistake: The y-intercept is wherever the line crosses an axis.
The y-intercept is where it crosses the y-axis, at x = 0. Crossing the x-axis gives the x-intercept.