Module 5.1

Understanding linear functions

What is a linear function?

P. 199

A race car can travel up to 210 mph. If x was the independent variable – the number of hours it travels –And y was the dependent variable – the number of miles it travels –The function that would represent the number of miles the car travels –if it could travel continuously at this speed – is 𝒚 = 𝟐𝟏𝟎𝒙Solutions are shown in the graph below:

P. 199

Fill in the table using the data points from the graph above.

Using the table, check that x has a constant change between consecutive terms.

Now check that y has a constant change between consecutive terms.

Using the answers from before, what change in x corresponds to a change in y?

______________________All linear functions behave similarly to this one.The graph of a linear function forms a straight line: LINEAR from the word “LINE”

A solution of a linear equation is any ordered pair that makes the equation true.

Is 𝒚 = 𝒙𝟐 a linear function?Here’s a table of values:

Does a constant change in x correspond to a constant change in y?

Here’s its graph:

Is it possible for a continuous linear function (graph) and a discrete linear function (graph) to share the same linear equation?

How about this example:

What’s the total cost of buying x suits priced at $210 each?𝒚 = 𝟐𝟏𝟎𝒙

What does its graph look like?Is it continuous or discrete?

Any linear function can be represented by a linear equation.

A linear equation is any equation that can be written in this form: 𝑨𝒙 + 𝑩𝒚 = 𝑪where A is the coefficient of x , B is the coefficient of y, and A, B, and C are real numbers.

This is called the Standard Form of a Linear Equation.

Examples:

This is the first of 3 forms you will learn.Another one, which you’ve likely heard of, is called Slope-Intercept form.

P. 202

If an equation is in standard form, what arethe exponents of x and y? Since the exponents are not visible, they must be 1.

Is 𝝅 an acceptable value for any of the coefficients in a linear equation?Yes; 𝝅 is a real number, so it can be a coefficient in a linear equation.

Equation A B C𝟐𝒙 + 𝟑𝒚 = 𝟏𝟐

𝒙 + 𝒚 = 𝟓

−𝟒𝒙 + 𝒚 = −𝟖

𝟖𝒙 = 𝟑𝟐

−𝒙 − 𝒚 = 𝟎

−𝟑𝒚 = −𝟗

𝟐𝒙 = 𝟖 − 𝟐𝒚

𝟒𝒙 = 𝟐𝒚

In order for an equation to be linear:

1) The exponents of x and y must be 1Problem equation: 𝟐𝒙𝟐 + 𝟑𝒚 = 𝟔

2) x and y can’t be multiplied togetherProblem equation: 𝟑𝒙𝒚 = 𝟔

3) x and y can’t be denominators, exponents, or in radical signs

Problem equations: 𝟒

𝒙+ 𝟐𝒚 = −𝟐

𝟑𝒙 − 𝒚 = 𝟎

−𝟐𝐱 +𝒚𝟖 = 𝟏𝟐

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𝒚 =

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