Investigating $y=mx+b$

The slope of a line

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 lower 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

What is the relationship between $m$ and the slope? Are they the same or are they different?

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

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

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.

Use the sliders to change the values of $m$ and $b$. What is the shape of the graph of $y=mx+b$ (triangle, circle, line, square, …)?

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

Look at the $y$-intercepts of the lines $y=2x+1$ and $y=4x+1$. Are they the same or are they different?

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

Look at the slopes of $y=2x+1$ and $y=2x+3$. Are they the same or are they different?

Set $m$ to 0 and use the slider for $b$ to change its value. Are lines with $m=0$ horizontal, vertical, or neither?

Each row of this table gives an equation for a line. Find the $y$-intercept of that line algebraically, by setting $x$ to $0$ in its equation.

equation for the line	set $x=0$	$y$-intercept

You can now see that:

The $y$-intercept of the line $y=mx+b$ is the point $(0, b)$.

Each row of this table gives an equation for a line. Find the slopes of these lines algebraically, by subtracting their $y$-values when $x=0$ and $x=1$.

equation for the line	$x$- and $y$- values	slope computation

As you can see:

The slope of the line $y=mx+b$ is $m$.

Finding an equation for a line

As you have learned, the numbers $m$ and $b$ in the equation $y=mx+b$ give the slope and $y$-intercept of the line that is the graph of that equation. Because of this:

$y=mx+b$ is called the slope-intercept form of the equation for a line.

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

Write equations in slope-intercept form 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?)

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

Write equations in slope-intercept form for two lines with slope 2.

Can you find equations for other lines with slope 2?

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

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

Write an equation in slope-intercept form 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?

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

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

What is an equation in slope-intercept form for the red line on the grid to the left? You can use the sliders and blue line to help you answer this question.

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

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

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 $(2,-3)$.