Investigating $y=mx$

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

Lines with positive slope

Question 1. 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. Then press the tab key to move to the next blank cell. 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 $y=2x$.

$x$	$y$
5
2
0
-2
-5

What do you think the graph of $y=2x$ will be?

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

Click on to graph the equations $y_1=x$ and $y=2x$ on the grid to the left.

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

Question 2. What do you think the graph of the equation $y=3x$ would look like if it were placed on the grid with these two lines?

Question 3. Click on to see the example for question 3. 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 bottom left portion of your screen to change the value of $m$ and answer the following questions.

What happens to the graph of $y=mx$ as $m$ increases from 0 to 5?

What happens to the graph of $y=mx$ as $m$ decreases?

What happens to the graph of $y=mx$ when $m=0$?

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 as you go 1 unit to the right.

Question 4. Use the slider and the graphs to the left to complete the table below. The equation for $y$ is shown under the grid to the left. Calculate each slope by looking at the graph.

equation	$m$	slope
$y=x$
$y=2x$
$y=3x$
$y=.5x$
$y=1.5x$

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

Lines with negative slope

Question 5. Click on to see the example for question 5. If $y=-x$, what will $y$ be if $x=5$? Enter this value in the $y$ column corresponding to $x=5$ in the table to the right. Repeat this for the other $x$ values in the table, using the equation $y=-x$.

$x$	$y$
5
2
0
-2
-5

What do you think the graph of $y=-x$ will look like?

How will it be similar to the graphs in the diagrams above? How will it be different from those graphs?

Question 6. Complete the table of values for $y_2=-2x$:

$x$	$y_2$
5
2
0
-2
-5

What do you think the graph of $y_2=-2x$ will be?

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

Question 7. Click on to graph the equations $y=-x$ and $y_2=-2x$ on the grid to the left.

Compare the graphs of $y$ and $y_2$. Describe what the graph of $y=-3x$ would look like on the same grid with these graphs.

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.

Putting it all together

Question 8. Click on to see the example for question 8.

What happens to the graph of $y=mx$ as you slide $m$ from 0 to −5?

What happens to the graph of $y$ as $m$ goes from 0 to 5?

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

What value does $m$ need to have for the graph of $y$ to match the graph of $y_2=-x$ (the red 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
$y=-3x$
$y=-.5x$
$y=3x$
$y=.5x$

What do you think the slope of $y=0.2x$ is?

Question 9. Describe any relationship you see between $m$ and the slope of $y=mx$.

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