Investigating $y=x+b$

Equations such as $y=x+2$ are called linear equations. In this lab you will study this and other similar equations, learn about their graphs, and learn why they are called linear.

Question 1. If $y=x$, what will $y$ be if $x=3$? Enter this value in the $y$ column corresponding to $x=3$ in the table to the right. Then press the tab key to move to the next blank cell. Notice that a new red point is added to the grid to the left, with coordinates given by the row you just completed. Repeat this for the other $x$ values in the table, using the equation $y=x$.

$x$	$y$
3
2
1	1
0
-1
-2	-2
-3

What do you notice about all of the red points on the grid?

Click on to graph the equation $y=x$ on the grid to the left. If there are any points that aren't on the line, you have made a mistake in the table. Correct any mistakes you find so that all of the points are on the line.

Question 2. Click on to see the example for question 2. The red line is the graph of $y_1=x$. Complete the table of values for the equation $y=x+2$:

$x$	$y$
3
2
1	3
0
-1
-2	0
-3

What do you notice about all of the blue points on the grid? How do your results compare to those from the first example? (Remember that the red line is the graph of the equation in the first example.)

Click on to look at the graphs of $y_1=x$ and $y=x+2$. Correct any mistakes you have made in the table so that all of the blue points are on the blue line.

Question 3. Click on to see the example for question 3. The red line is the graph of $y_1=x$ and the blue line is the graph of $y_2=x+2$. Complete the table of values for the equation $y=x-2$:

$x$	$y$
3
2
1	-1
0
-1
-2	-4
-3

What do you notice about all of the green points on the grid?

Click on to see the graphs of $y_1=x$, $y_2=x+2$, and $y=x-2$. Correct any mistakes you have made in the table so that all of the green points are on the green line.

Question 4. Why do you think that $y_1=x$, $y_2=x+2$, and $y=x-2$ are called linear equations?

The $y$-intercept of a line

Question 5. For each of the three equations we have graphed, find the point where that equation's graph crosses the $y$-axis (the vertical axis with the arrow at the top). What are the $(x,y)$ coordinates of that point for

$y_1=x$ (the red line)?
$y_2=x+2$ (the blue line)?
$y=x-2$ (the green line)?

The point where a line crosses the vertical axis or $y$-axis is commonly called the $y$-intercept of the line.

Question 6. Click on to see the example for question 6. The blue line is the graph of $y_1=x+4$, the purple line is the graph of $y=x$, and the green line is the graph of $y_3=x-4$. Use the slider in the bottom left portion of your screen to change the value of $b$ in the equation $y=x+b$. Click and drag the slider’s “handle,” and then use the left and right arrow keys to make small changes.

What value should $b$ have for the graph of $y=x+b$ to match the graph of $y_1=x+4$ (the blue line)?

What value should $b$ have for the graph of $y$ to match the graph of $y_3$?

Use the slider to change the value of $b$ and complete the table below. The equation for $y$ is shown below the grid.

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

Describe the relationship between the value of $b$ and the $y$-intercept of each line.

What do you think the $y$-intercept of the graph of $y=x+10$ is?

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