Investigating $y=mx+b$

The slope and $y$-intercept of a line help to describe it. In this lab you will look at equations such as $y=3x+2$ and their graphs, and learn about each graph's shape, slope, and $y$-intercept.

The slope of a line

Question 1. The slope of a line measures how steep the line is. It is the amount of change in the height of a line as you go 1 unit to the right. If the height of a line decreases as you move to the right, its slope is negative.

Use the slider in the bottom left portion of your screen to change the value of $m$ in the equation $y=mx$ and complete the table below. Use the grid to the left to determine each slope. The equation for $y$ is shown below the grid.

$m$	equation for line	slope
2	$y=2x$	2
3
-2
-1

What is the relationship between $m$ and the slope?

Slope measures the steepness of a line. What do you think the slope of a horizontal line is? Use your answer to the previous question and the slider to check your answer.

The $y$-intercept of a line

Question 2. Click on to see the example for question 2. The $y$-intercept of a line is the point where it crosses the $y$-axis. Use the slider to change the value of $b$ in the equation $y=x+b$ and complete the table below.

$b$	equation for line	$y$-intercept
2	$y=x+2$	$(0,2)$
3
-2
-1

What is the relationship between $b$ and the $y$-intercept?

What would an equation for this line be if its $y$-intercept were $(0,0)$?

The slope and $y$-intercept of lines in the form $y=mx+b$

So far you have learned that the slope of the line $y=3x$ is 3 and that the $y$-intercept of the line $y=x+2$ is $(0,2)$, but what about the slope and $y$-intercept of $y=3x+2$? Is the graph of $y=3x+2$ a straight line? In this section we will answer these questions.

Question 3. Click on to see the example for question 3. Use the sliders to change the values of $m$ and $b$. What is the shape of the graph of $y=mx+b$?

Use the slider to change the value of $m$ without changing $b$. Does changing $m$ affect the $y$-intercept of a line?

What can you say about the $y$-intercepts of the lines $y=2x+1$ and $y=4x+1$?

Use the slider to change the value of $b$ without changing $m$. Does changing $b$ affect the slope of a line?

What can you say about the slopes of $y=2x+1$ and $y=2x+3$?

Set $m$ to 0 and use the slider for $b$ to change its value. What kind of lines do you get when $m=0$? Why?

Finding an equation for a line

Question 4. Use the sliders to change the values of $m$ and $b$ and answer the following questions.

Write equations for two lines with $y$-intercept $(0,3)$.

Can you find equations for other lines with $y$-intercept $(0,3)$? (yes or no)

Can you find an equation for a specific line if all you know is its $y$-intercept? (Can you be sure that you have the right line?)

Question 5. Use the sliders to change $m$ and $b$ and answer the following questions.

Write equations for two lines with slope 2.

Can you find equations for other lines with slope 2? (yes or no)

Can you find an equation for a specific line if you only know its slope?

Question 6. Use the sliders to change $m$ and $b$ and answer the following questions.

Write an equation for a line with slope 2 and $y$-intercept $(0,3)$.

How many lines have slope 2 and $y$-intercept $(0,3)$?

To find an equation for a line, is it enough to know only its slope and $y$-intercept?

Question 7. Use the sliders to change the values of $m$ and $b$ and complete the table below.

slope	$y$-intercept	$m$	$b$
-3	$(0,2)$
		1	-2
-1			3

Question 8. What is an equation for the line in the figure below? You can use the sliders to help you answer this question.

Graph

What is an equation for the line with slope −2 that has the same $y$-intercept as the line in the figure above?

What is an equation for the line with $y$-intercept $(0,-2)$ that has the same slope as the line in the figure above?

Question 9. Click on to see the example for question 9. The points $(0,1)$ and $(2,-3)$ are shown on the grid. Use the sliders to answer the following questions.

Find an equation for a line that goes through $(0,1)$ and $(2,-3)$.

Can you find another line that goes through these two points? (yes or no)

Find an equation for the horizontal line that goes through $(0,1)$.

Find an equation for the horizontal line that goes through $(2,-3)$.

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