Linear Functions

Poll

The three lines show a person’s distance from a park entrance as a function of time in seconds.

Who is walking the fastest?

• Person A

• Person B

• Person C

Slope and Rate of Change

When you compared the three lines to see who was the fastest, you were comparing the slopes of the lines, or the rate of change.

In other words, you weren't just looking at the change in time, or the change in distance. You were looking at how they compared to each other.

Add to the DEFINITIONS section of your digital notebook

• Rate of change: how fast one variable (the output or y) is changing compared to another variable (the input or x); the change in y divided by the change in x

• Constant rate of change: a function has a constant rate of change if the rate of change between any two points is always the same

• Slope: a way to measure the steepness and direction of the line; another way of talking about rate of change

Dropdown

The total amount of money spent on T-shirts is a function of the number of T-shirts bought. One T-shirt costs $10.



That means 2 T-shirts cost ​$20, 5 T-shirts cost $50 , and 100 T-shirts cost ​$1000.



To find these values, we ​

Select answer

the ​

Select answer

by the ​

Select answer

Fill in the Blanks

Given that

total cost = cost of 1 T-shirt $\times$ number of T-shirts,

write an expression for the cost of x T-shirts

Open Ended

Let's add something new to the story problem.

In addition to the cost per shirt, there is a $20 shipping fee to get your shirts. The fee is the same no matter how many shirts you buy.

How will this affect the total cost for x T-shirts?

Drag and Drop

Write an equation for the total cost, y, of x T-shirts.



y = ​

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x + ​

Placeholder

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10

20

You just wrote a linear function!

A linear function is a function with a constant rate of change. It is called "linear" because on a graph, it is a straight line.

Linear functions never have exponents. The form we will work with is called slope-intercept form, because it shows us the slope and y-intercept.

Add to the DEFINITIONS section of your digital notebook

• Linear function: a function with a constant rate of change; the graph of a linear function is a straight line

• Slope-intercept form: a way of writing a linear function that shows the slope and y-intercept of the function

Add to the FORMULAS section of your digital notebook

Example: Writing a Linear Function from a Graph

Drag and Drop

We find slope with the formula $$m=\frac{y_2-y_1}{x_2-x_1}$$ . In other words, we find the change in y, and divide by the change in x.



The two points we picked were $$\left(x_1,\ y_1\right)=\left(-4,\ 0\right)$$ and $$\left(x_2,\ y_2\right)=\left(0,\ 2\right)$$

So the change in y will be $$y_2-y_1=$$ ​

Placeholder

-​

Placeholder

= ​

Placeholder

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2

0

Drag and Drop

We find slope with the formula $$m=\frac{y_2-y_1}{x_2-x_1}$$ . In other words, we find the change in y, and divide by the change in x.



The two points we picked were $$\left(x_1,\ y_1\right)=\left(-4,\ 0\right)$$ and $$\left(x_2,\ y_2\right)=\left(0,\ 2\right)$$

So the change in x will be $$x_2-x_1=$$ ​ ​

Placeholder

-​ ​

Placeholder

= ​

Placeholder

Drag these tiles and drop them in the correct blank above

0

-4

4

Multiple Choice

The change in y is 2 and the change in x is 4. Since slope is $$\frac{change\ in\ y}{change\ in\ x}$$ , the slope is

• 4

• 2

• 1/2

Step 2: Find the y-intercept (b)

The y-intercept is the point where the line crosses the y-axis.

​

Example: Writing a Linear Function from a Graph

Multiple Choice

What is the y-intercept of this function?

• (0, 2)

• (2, 0)

• (0, -4)

• (-4, 0)

Fill in the Blanks

The y-intercept is the point (0, 2).

When we write the function, we will only care about the y-coordinate (b in slope-intercept form).

b = ?

Drag and Drop

Now we know the slope (1/2) and the y-intercept (2) of the function. Fill in the blanks to finish the function.



y = mx + b

y = ​

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x + ​

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1/2

2

Example: Writing a Linear Function from a Word Problem

Ms. Shier is on page 152 of a 700 page book. She reads about 100 pages per hour. Write a linear function to describe the number of pages, y, that Ms. Shier has read after x hours.

Step 1: Find the slope (m)

Often, the slope is given to you. Look for a speed, a price per item, the word “rate,” or units with “per” or division (like miles per hour or

m/s)

Multiple Choice

Ms. Shier is on page 152 of a 700 page book. She reads about 100 pages per hour. Write a linear function to describe the number of pages, y, that Ms. Shier has read after x hours.

What is the slope (rate of change)?

• 152

• 700

• 100

Example: Writing a Linear Function from a Word Problem

Ms. Shier is on page 152 of a 700 page book. She reads about 100 pages per hour. Write a linear function to describe the number of pages, y, that Ms. Shier has read after x hours.

Step 2: Find the y-intercept (b)

The y-intercept tells us the output when the input is zero. In other words, the y-intercept is the starting value of the output.

Multiple Choice

Ms. Shier is on page 152 of a 700 page book. She reads about 100 pages per hour. Write a linear function to describe the number of pages, y, that Ms. Shier has read after x hours.

What is the y-intercept (starting value)?

• 152

• 700

• 100

Drag and Drop

Now we know the slope (100) and the y-intercept (152) of the function. Fill in the blanks to finish the function.



y = mx + b

y = ​

Placeholder

x + ​

Placeholder

Drag these tiles and drop them in the correct blank above

100

152