Investigating $y=mx$

The slope of a line is a number that measures how steep the line is. In this lesson you will learn about the meaning of slope, and find the slopes of several linear graphs.


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.

Slope 1 Slope 2

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.

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.

Slope = −1

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 $m$.