Finding Equations for Lines

You know that 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 move 1 unit to the right. In this lab you will learn another method for finding the slope of a line. You will also learn how to find an equation for a line if you know its slope and a point on the line, or two points on the line.

You may need a pencil and scratch paper to complete this lab.

Slope as rate of change

Question 1. Jane is descending a 20-foot rock. She climbs down the rock at a rate of 2 feet per minute (so, each minute her height decreases by 2 feet).

In this example $t$ represents time in minutes and $h$ represents Jane's height. When $t=0$, $h=20$ ft. What is $h$ when $t=1$? Enter the value you just found in the $h$ column corresponding to $t=1$ in the table to the right. Repeat this for the other $t$ values in the table to chart her progress. (The points representing Jane's height are graphed on the grid to the left.)
$t$$h$
020
1
216
3
4
5
6

What do you notice about all of the points on the graph?

Click on . What is the slope of the red line on the grid to the left? (Remember that slope is the amount of change in the height of a line as you move 1 unit to the right.)

We say that the rate of change between the points $(0,20)$ and $(2,16)$ is

$$\text"change in height"/\text"change in time" = {16-20}/{2-0} = {-4}/2 = -2$$

Find the rate of change between the points shown in this table. All of these points are on the graph.
Point 1Point 2rate of change
$(5,10)$$(2,16)$
$(4,12)$$(5,10)$

What do you notice about the rate of change?

What is the relationship between the value you found for slope and the values you found for rate of change?
The slope of the line through the points $(x_1,y_1)$ and $(x_2,y_2)$ is $${y_2-y_1}/{x_2-x_1}$$

Finding an equation for a line given its slope and $y$-intercept

Question 2. Click on to see the graph of $y=2x+1$. 2 is the slope of the line and 1 indicates where the line crosses the $y$-axis. This $y=mx+b$ form of the equation for a line is called the slope-intercept form. Use the sliders to change the values of $m$ and $b$ and notice how they affect the graph. Complete this table by using the sliders.
equationslope$y$-intercept
$y=3x+1$ $(0,1)$
$y=x-4$
$y=-2x-1$

Question 3. Complete this table. Use the sliders to check your answers.
slope$y$-interceptequation
3$(0,4)$
2$(0,1)$
−0.5$(0,-2)$

Finding equations for lines with a given point and slope

In the last section you found an equation for a line given its slope and a particular point on the line, its $y$-intercept. In this section you will learn how to find an equation for a line given its slope and any point on the line, not necessarily the $y$-intercept.

Question 4. Click on to see the example for question 4. Use the sliders to change the values of $m$, $x_1$, and $y_1$ and notice how they affect the graph of $y-y_1=m(x-x_1)$. The equation for the line is always shown below the grid. Use the sliders to complete this table.

$m$$x_1$$y_1$pointequation
213(1, ) $y-3=2(x-1)$
312(1, )
−521(2, )

What do you notice about the relationship between the values of $x_1$ and $y_1$ and the point on the line?

Question 5. Click on to see the example for question 5. The red line is the graph of $w-4=2(x-3)$. Find a point other than $(3,4)$ that is on the red line.

Slide $m$ to the value 2. Next, slide $x_1$ to the value of the $x$-coordinate of your new point and $y_1$ to the value of the $y$-coordinate of your new point. What is the equation (in $x$ and $y$) for the blue line?

Is the graph different from the graph of $w-4=2(x-3)$ (the red line)?

Find another point on the line.

Slide $x_1$ and $y_1$ as before using the new point. Is the graph different?

What is the equation for the line you found using this new point?

If a line contains the point $(x_1,y_1)$ and has slope $m$, then its equation can be written as $y-y_1=m(x-x_1)$.
$y-y_1=m(x-x_1)$ is called the point-slope form of the equation for a line.

Question 6. We can write the equation $y-4=2(x-3)$ in slope-intercept form:

Equation: $y-4=2(x-3)$
Distribute: $y-4=2x-6$
So, $y=2x-6+4$
  $y=2x-2$

Rewrite the two equations you found in Question 5 in slope-intercept form.

What do you notice about these two equations?

Finding equations for lines through two points

Question 7. Click on to see the example for question 7. Sometimes we don't know the slope of a line, but we do know two points on the line. For example, you might be told that the price of 4 baskets of strawberries is \$6 and the price of 2 baskets of strawberries is \$3. This means that you have two points, $(4,6)$ and $(2,3)$. To find an equation for the line through these points you need to find the slope first.

What is the slope between the points $(2,3)$ and $(4,6)$?

Use the point $(4,6)$ and the slope to find an equation for the line in point-slope form.

Slide the values of $m$, $x_1$, and $y_1$ in the equation $y-y_1=m(x-x_1)$ to get the equation you just found. Does its graph pass through the point $(2,3)$?

Write the equation you found in slope-intercept form.

Question 8. Write an equation for the line that passes through the points $(3,1)$ and $(4,-1)$ in slope-intercept form.

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