Investigating $y=x+b$

If $y=x$, what will $y$ be if $x=2$? Enter this value in the $y$ column corresponding to $x=2$ in the table to the right. Then press the tab key, or tap on a touch screen, 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 $y=x$.

$x$ $y$

What shape is being formed by all of the red points on the grid (triangle, circle, line, square, …)?

Click 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.

The red line is the graph of $y_1=x$. Complete the table of values for the equation $y=x+2$.

$x$ $y$

Look at the blue points on the grid. Do they form the same shape as in yeqx or a different shape? (Remember that the red line is the graph of the equation from yeqx.)

Is the shape formed by the blue points above or below the red line?

Click 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.

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 $y=x-2$.

$x$ $y$

Look at the green points on the grid. Do they form the same shape as the points in the previous two questions, or a different shape?

Is the shape formed by the green points above both the red line and the blue line, below both the red line and the blue line, or between the red line and the blue line?

Click 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.

This is why the equations $y_1=x$, $y_2=x+2$, and $y=x-2$ are called linear equations: because their graphs are lines.

The $y$-intercept of a line

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.

The blue line is the graph of $y_1=x+4$, the red line is the graph of $y_2=x$, and the green line is the graph of $y_3=x-4$. Use the slider in the lower left portion of your screen to change the value of $b$ in the equation $y=x+b$ (the purple line).

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

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

The $y$-axis is the place where $x=0$. So you can also find the $y$-intercept of a line algebraically: by setting $x=0$ in the equation for that line.

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

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

Based on these last two questions, you can see that:

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