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6.4 Lines of the form y = mx + b

The equation  y = mx is the standard equation of a line passing through the origin.  For most travel problems, starting at the origin means starting at time = 0 and the starting position = 0.  In sports, competitors often start races at locations other than the zero point.  For example in track relays, runners will often receive their batons at different times and at different locations.  On the football field, a foot race can start anywhere at any time and travel in any direction.

Example 6.5:  Tony catches a pass at his team's 18 yard line and runs down the field at 8 yards per second.  Plot his travel.  If he's not stopped, how long will it take him to cross the 50 yard line? 

Solution:  In word problems, it is always helpful to draw a picture to help visualize the problem. 

As with the other problems, we'll say that time is on the x-axis and position is on the y-axis.  To plot his travel, we need to find an equation that shows his position as a function of time.  Since the problem doesn't give us a starting time, we'll use the time he catches the pass as the starting time (x = 0).   The distance he'll travel every second after the catch is 8x (8 yards/sec. x x seconds).  Since he started at the 18 yard line, his position at time x is given by the following equation:

                        y = 8x + 18      where   y = field position, in yards

                                                            x = time elapsed since the catch, in seconds

We can plot the graph by finding two points that fit the equation and drawing a line through them.  We know that he's at the 18 yard line at time = 0 so,

y = 18    at        x = 0

Therefore, (0, 18) is one point on the line.

At x = 3 seconds,         y= 8(3) + 18

= 24 + 18 = 42                   

Therefore the point (3,42) is also on the line.

By plotting the points and connecting them with a line, we get the line that represents the equation.

To find when Tony will cross the 50 yard line, we take his position at that time, 50, and exchange it with the y in the equation.  Then we solve the equation for x.

If we look at the graph, we can see that Tony crosses the 50 yard line when x = 4 sec.

Note that in this problem, the axes were graduated differently.  The distance of 10 yards in the y-direction is about the same as the distance of 2 seconds in the x-direction.  If the axes had the same graduation, the slope of 8 would look much steeper.

Also note that this problem only looks at a section of the line y = 8x + 18. The graph shows only where x is greater than 0 and y is less than 50.  The equation, y = 8x + 18, really also applies to numbers outside of this range.  When a line on a graph is not limited by a range XE  "range" , arrows are usually placed on the ends of the lines to show that the line continues.  The range shown in Figure 6.5 shows that the line segment only represents a section of the full line.

Example 6.6:  Two plays later, Tony catches the ball on the on the 28 yard line on the opponent's side of the field.  On the opponent's side, of course, the yard markers are decreasing.  He still runs at 8 yards per second.  Plot his position with zero as the time he catches the ball.  If he isn't stopped, how many seconds will it take him to cross the goal line?

Solution:  Since he's now on the opponent's side of the field, the yard markers are decreasing.  We can visualize the problem better if we show the yardage markers to increase to the right like the number line.  In other words, we could view the problem from the other side of the field.

He starts at the 28 yard line.  Every second he'll be 8 yards closer to the goal line.  His position will therefore decrease by 8 yards per second. mHis position will be:

y = -8x + 28    where   y = field position, in yards

x = time elapsed since the catch, in seconds

Again, we can plot a line by finding two points.   We know he's at the 28 yard line (y = 28) at the start (x = 0).  So (0,28) is one point. 

At        x = 2                y = -8(2) + 28

 = -16 + 28 = 12                   Plot (2,12)

When Tony crosses the goal line, y = 0.  So:

Tony crosses the goal line 3.5 seconds after catching the ball.

Sincem, -8, is negative in this equation, the line y = -8x + 28 has a negative slope.  The examples show that lines that run up to the right and down to the left have a positive slope XE  "positive slope" .  Lines that go up to the left and down to the right have a negative slope.  For lines that are perfectly horizontal, the y values do not change .  Therefore:

Horizontal lines have a zero slope XE  "zero slope" .  For vertical lines, the line does not change in the x direction .  Therefore:

The value of m therefore becomes an unreal number.  Vertical lines are said to have no slope.