Investigating $y=mx$

Lines with positive slope

If $y=2x$, what will $y$ be if $x=5$? Enter this value in the $y$ column corresponding to $x=5$ in the table to the right. Notice that a blue 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 $y=2x$. (The equation $y_1=x$, which you learned about in the last lesson, is already graphed on the grid.)

$x$ $y$

What shape do you think the graph of $y=2x$ will be (triangle, circle, line, square, …)?

Which points are higher when $x$ is positive, the red points from $y_1=x$ or the blue points from $y=2x$?

Click to graph the equation $y=2x$ on the grid to the left.

Does the graph of $y=2x$ look like you thought it would?

Think about what the graph of the equation $y=3x$ would look like if it were placed on the grid with these two lines. When $x$ is positive, would it be higher than both lines, lower than both lines, or between the two lines?

The slider for $m$ is set to 3 in the equation $y=mx$, so the equation for the green line is $y=3x$. Does the graph of $y=3x$ appear as you guessed it would?

Use the slider in the lower left portion of your screen to change the value of $m$ and answer the following questions.

Look at the graph of $y=mx$ as you slide $m$ from 0 to 5. Does the graph become steeper or less steep?

Look at the graph of $y=mx$ as $m$ decreases from 5 to 0. Does it become steeper or less steep?

When $m=0$, is the graph of $y=mx$ horizontal, vertical, or neither?

What value does $m$ need to have for the graph of $y$ to match the graph of $y_1=x$?

What value does $m$ need to have for the graph of $y$ to match the graph of $y_2=2x$?

When you slide $m$ the graph of $y=mx$ changes in steepness. The measure of this steepness is called the slope of the line.

The slope of a line is the amount of change in the height of the line each time you go 1 unit to the right.

Use the slider and the graphs to the left to complete the table to the right. The equation for $y$ is shown below the grid to the left. Calculate each slope by looking at the graph.

equation $m$ slope

What do you think the slope of the line is when $m=10$?

Lines with negative slope

If $y_1=-x$, what will $y_1$ be if $x=5$? Enter this value in the $y_1$ column corresponding to $x=5$ in the table to the right. Repeat this for the other $x$ values in the table, using the equation $y_1=-x$.

$x$ $y_1$

Think about the graph of $y_1=-x$. Will it have the same shape as the graphs from earlier in the lesson, or a different shape?

Earlier in this lesson, as $x$ moved to the right, the plotted values moved up. As $x$ moves to the right on this graph, do the $y_1$-values move up or down?

Complete the table of values for $y_2=-2x$:

$x$ $y_2$

Will the graph of $y_2=-2x$ have the same shape as the graphs from earlier in the lesson, or a different shape?

Which points are higher when $x$ is positive, the blue points from $y_2=-2x$ or the red points from $y_1=-x$?

The equations $y_1=-x$ and $y_2=-2x$ are graphed on the grid to the left.

If you graphed $y=-3x$, would it have the same shape as the graphs from earlier in the lesson, or a different shape?

When $x$ is positive, would the graph of $y=-3x$ be higher than the graphs of $y_1=-x$ and $y_2=-2x$, lower than the graphs of $y_1=-x$ and $y_2=-2x$, or between the graphs of $y_1=-x$ and $y_2=-2x$?

Give an equation for a line with a graph that would fall between the red and the blue lines. (Hint: In $y=mx$, what could $m$ be?)

The line shown below has slope $-1$ because as the $x$-coordinate is increased by 1 you must move down 1 (in the $-1$ direction) to return to the line.

Putting it all together

What happens to the graph of $y=mx$ as you slide $m$ from 0 to $-5$? Does it become steeper or less steep?

What happens to the graph of $y$ as $m$ goes from 0 to 5? Does it become steeper or less steep?

What value does $m$ need to have for the graph of $y$ to match the graph of $y_1=x$ (the red line)?

What value does $m$ need to have for the graph of $y$ to match the graph of $y_2=-x$ (the blue line)?

Use the slider and the graph to the left to complete the table below. Use the grid to determine the slope of the green line.

equation	$m$	slope

What do you think the slope of $$y={1}/{5}x$$ is?

You don’t need to see the graph of an equation to find its slope. Instead, you can just use the equation to determine how much $y$ changes when you increase $x$ by 1.

In each row of the table, find the slope of $y=2x$ by subtracting its $y$-values at the two indicated points, where $x$ increases by 1. The line, and your two points, will be plotted on the grid to the left.

	$x$- and $y$- values	slope computation

Notice that you can calculate the slope starting anywhere on the line, and you’ll get the same answer.

Because you get the same slope for a straight line wherever you start from, you might as well pick simple $x$-values to use. The simplest $x$-values are almost always $x=0$ and $x=1$.

Find the slopes of these lines 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 from the last few questions:

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