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Lesson 13: Linear and nonlinear functions

# Linear equations and functions: FAQ

## What is a proportional relationship?

A proportional relationship is when two quantities change so that they maintain a constant ratio to one another. For example, if y is directly proportional to x, then if x doubles, y will also double. And if x triples, y will also triple, and so on.

## What do we mean by "solutions" to linear equations?

Solutions to linear equations are the x and y values that make the equation true. For example, left parenthesis, 2, comma, 7, right parenthesis is a solution to the equation y, equals, 3, x, plus, 1 because when we substitute 2 in for x and 7 in for y, we get a true statement: 7, equals, 3, left parenthesis, 2, right parenthesis, plus, 1.

## What is slope?

Slope is a measure of the steepness of a line. It is calculated by dividing the rise (change in y) by the run (change in x) between two points on the line.

## What are intercepts?

Intercepts are the points at which a line crosses the x- or y-axis. The x-intercept is the point at which the line crosses the x-axis, and the y-intercept is the point at which the line crosses the y-axis.

## What is slope-intercept form?

Slope-intercept form is a way of writing a linear equation. It has the form y, equals, m, x, plus, b, where m is the slope and b is the y-intercept.

## How do we graph a line in slope-intercept form?

To graph a line in slope-intercept form, we can start by plotting the y-intercept, which is given by the b value in the equation. From there, we can use the slope (the m value) to find more points on the line.

## How do we write an equation in slope-intercept form?

To write an equation in slope-intercept form, we need to find the slope and the y-intercept of the line. We can use two points on the line to calculate the slope, and we can use the slope and one of the points to solve for the y-intercept.

## What is a function?

A function is a mathematical rule that matches inputs to outputs. We can think of it like a machine: put a number in, the machine does some calculations, and out pops a corresponding number.
Inputs and outputs don't have to be numbers. Functions themselves can be inputs and outputs.

## What is a linear function?

A linear function is a type of function that produces a straight line when graphed. It follows the form y, equals, m, x, plus, b, where m is the slope and b is the y-intercept.

## How do we compare two linear functions?

We can compare two linear functions by looking at their slopes and y-intercepts. If the slopes are the same, the lines are parallel. If the slopes are different, the lines will intersect at some point. We can also compare the y-intercepts to determine where the lines cross the y-axis.

## How do we construct a linear model for a real-world relationship?

To construct a linear model, we need to determine the slope and y-intercept of the relationship. We can use two data points to find the slope, and then use one of the data points to solve for the y-intercept.

## What are some real-world applications of linear functions?

Linear functions can be used to model many real-world relationships. For example, a company might use a linear function to predict future sales based on past performance. A scientist might use a linear function to model the relationship between two variables in an experiment. Linear functions can also be used in finance to calculate interest rates or investment returns.

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