This algebra 1 linear functions lesson explains linear functions: slope-intercept, point-slope and standard form in one printable page, with definitions, a method and a worked example.
A linear function has a constant rate of change called its slope. It can be written as y = mx + b (slope-intercept), y – y1 = m(x – x1) (point-slope) or Ax + By = C (standard). Find the slope from two points, graph the line from any form, and write models from tables, graphs or words.
Algebra 1 Linear Functions Lesson: key facts
- Standards F-IF.C.7a, F-LE.A.1a, F-LE.A.2 and A-CED.A.2: graph linear functions, recognize constant rates of change, and write linear models.
- Slope from two points: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
- Slope-intercept \(y=mx+b\), point-slope \(y-y_{1}=m(x-x_{1})\), standard \(Ax+By=C\).
- Parallel lines have equal slopes; perpendicular lines have slopes whose product is \(-1\).
- In a model, the slope is the rate of change and the y-intercept is the starting value.
What is slope and how do you find it?
A linear function changes by the same amount every time \(x\) goes up by 1. That constant rate of change is the slope. Given two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\), subtract the coordinates in the same order:
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
For \((-2,1)\) and \((4,5)\), \(m=\frac{5-1}{4-(-2)}=\frac{4}{6}=\frac{2}{3}\). A positive slope rises from left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has undefined slope.
What are the three forms of a linear equation?
Slope-intercept form: \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.
Point-slope form: \(y-y_{1}=m(x-x_{1})\), for a line with slope \(m\) through \((x_{1},y_{1})\).
Standard form: \(Ax+By=C\), usually with integer coefficients and \(A\ge 0\).
To convert, solve for \(y\) or clear fractions. For example, \(2x+3y=12\) becomes \(3y=-2x+12\), so \(y=-\frac{2}{3}x+4\).
How do you graph a line from its equation?
- Write the equation in a form that shows a point and the slope.
- Plot that point, such as the y-intercept \((0,b)\).
- Use rise over run to plot at least one more point, then draw the line through them.
- Check a third point in the equation.
Worked example. Graph \(y=-\frac{2}{3}x+4\). Plot \((0,4)\). The slope \(-\frac{2}{3}\) means down 2 and right 3, which gives \((3,2)\) and \((6,0)\). Check: \(-\frac{2}{3}(6)+4=0\).
How are parallel and perpendicular lines related?
Parallel lines have equal slopes and different y-intercepts. Perpendicular lines have slopes that are negative reciprocals, so their product is \(-1\). If \(m=2\), a perpendicular line has slope \(-\frac{1}{2}\).
How do you write a linear model from a situation?
Find the constant rate of change (the slope) and the starting value (the y-intercept). A gym that charges a \(\$25\) sign-up fee plus \(\$18\) per month has cost \(C=18m+25\). Two points from a table also work: find \(m\) first, then use point-slope form.
Linear Functions: Slope-Intercept, Point-Slope and Standard Form: frequently asked questions
How do you find the slope of a line from two points?
Divide the change in y by the change in x, keeping the same point first in both differences. For \((1,3)\) and \((5,11)\), \(m=\frac{11-3}{5-1}=2\).
Which form of a linear equation should I use?
Use slope-intercept form to graph quickly from the y-intercept and slope. Use point-slope form when you know a point and the slope. Use standard form to find intercepts or to write equations with integer coefficients.
How do you convert standard form to slope-intercept form?
Solve the equation for \(y\). For \(3x+4y=12\), subtract \(3x\) to get \(4y=-3x+12\), then divide by 4 to get \(y=-\frac{3}{4}x+3\).
How do you write a line parallel or perpendicular to a given line?
A parallel line keeps the same slope. A perpendicular line uses the negative reciprocal of the slope. Then plug the given point into point-slope form.
What do slope and y-intercept mean in a word problem?
The slope is the amount the output changes per one unit of input, such as dollars per month. The y-intercept is the starting amount when the input is 0, such as a sign-up fee.
How to use this algebra 1 linear functions lesson
Use this algebra 1 linear functions lesson before the worksheet, then check what you learned with the quiz.
- Linear Functions: Slope-Intercept, Point-Slope and Standard Form worksheet (22 problems)
- Linear Functions: Slope-Intercept, Point-Slope and Standard Form quiz (10 questions)
- Linear Functions: Slope-Intercept, Point-Slope and Standard Form test (45 minutes)
- Previous lesson: Key Features of Graphs and Rate of Change
- Next lesson: Arithmetic and Geometric Sequences
See every chapter on the Algebra 1 math page.