## Key Equations

slope-intercept form of a line | [latex]f\left(x\right)=mx+b\\[/latex] |

slope | [latex]m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\[/latex] |

point-slope form of a line | [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)\\[/latex] |

## Key Concepts

- The ordered pairs given by a linear function represent points on a line.
- Linear functions can be represented in words, function notation, tabular form, and graphical form.
- The rate of change of a linear function is also known as the slope.
- An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
- The initial value, or
*y*-intercept, is the output value when the input of a linear function is zero. It is the*y*-value of the point at which the line crosses the*y*-axis. - An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
- A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
- A constant linear function results in a graph that is a horizontal line.
- Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.
- The slope of a linear function can be calculated by dividing the difference between
*y*-values by the difference in corresponding*x*-values of any two points on the line. - The slope and initial value can be determined given a graph or any two points on the line.
- One type of function notation is the slope-intercept form of an equation.
- The point-slope form is useful for finding a linear equation when given the slope of a line and one point.
- The point-slope form is also convenient for finding a linear equation when given two points through which a line passes.
- The equation for a linear function can be written if the slope
*m*and initial value*b*are known. - A linear function can be used to solve real-world problems.
- A linear function can be written from tabular form.

## Glossary

- decreasing linear function
- a function with a negative slope: If [latex]f\left(x\right)=mx+b, \text{then} m<0\\[/latex].

- increasing linear function
- a function with a positive slope: If [latex]f\left(x\right)=mx+b, \text{then} m>0\\[/latex].

- linear function
- a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line

- point-slope form
- the equation for a line that represents a linear function of the form [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)\\[/latex]

- slope
- the ratio of the change in output values to the change in input values; a measure of the steepness of a line

- slope-intercept form
- the equation for a line that represents a linear function in the form [latex]f\left(x\right)=mx+b\\[/latex]

*y*-intercept- the value of a function when the input value is zero; also known as initial value