The graph of any equation y = mx + b, where m and b are
specific numbers , is a straight line. Equations of this form
are called linear equations .
Example: Find some points on the graph of y = 2x +
1.
Solution : First compute a table of values which satisfy
the given equation.
| x |
-2 |
-1 |
0 |
1 |
2 |
| y |
-3 |
-1 |
1 |
3 |
5 |
Plotting these pairs gives the graph shown in
Figure A8.
The graph of y = 2x + 1 is the straight line through these points.
Problems:
18. Sketch the graph of y = x2 + 1.
19. Sketch the graph of
(Consider x = -1.)
20. Sketch the graph of
(Consider x = -1.)
21. Sketch the graph of y = x2 - 2x + 1.
22. Sketch the graph of x = -y2 + 2y - 1.
LINES IN THE PLANE
In plane geometry, you learn that two distinct points lie
on exactly one straight line. Its slope
characterizes the direction of a line. Given any two points on a line, the slope
of the line is defined by
According to this definition, the slope of a horizontal line is zero
(0) and the slope of a vertical line is undefined.
Example: Consider the line joining (2, 3) and
(7, 5). The slope of this line is
Graphically we have Figure A9.
Problems:
23. What is the slope of the line joining (3, 5) and (7, 3)?
24. What is the slope of the line joining (2, 3) and (2 + h, 3 + 4h + h2)?
Through a given point P = (x, y), there is exactly one line with slope m. If the
coordinates of P and the
slope m are known, we can easily find an equation relating the x- and
y-coordinates of any point (x, y)
on the line. In general, the line through P = (x0, y0) with slope m
consists of all points whose
coordinates (x, y) satisfy
which can be rewritten as y - y0 = m(x - x0 ).
This is called the point-slope form of a line. It
displays the coordinates of one point on the line and
the slope m of the line.
Example: An equation of a line is y -3 = 2(x + 4).
The line then passes through the point
( x0,y0 ) = ( -4,3) and has slope of m = 2.
Problems:
25. Find an equation of the line through ( -3,-4) with slope of 5/3.
26. Find the slope and the coordinates of a point on the line whose equation is
27. Find an equation of the line through (6 , 2) with slope 0.
28. Find an equation of the line through (5 , 3) and (5 , -11).
Now we consider another useful form of a line, the slope-intercept form.
Example: If the point-slope form of a line is y - 3
= 2(x - 4), then we can rewrite the equation as
y = 2x - 8 + 3 or y = 2x - 5.
In this form one can easily tell that the point (0, -5) is
on the line and that the slope,
given by the coefficient of x , is 2. More generally the slope- intercept form is
y = mx + b,
where m is the slope and (0 , b) is the point where the
line intercepts the y-axis.
Problems:
29. Write in slope-intercept form an equation for the line through (5, 1) with
slope -2.
30. Write 3x - 7y = 4 in the slope-intercept form.
31. What is the slope of the line 2x + 5y = 8? What is the y-intercept?
Example: The fol lowing three points lie on the same
line: ( -2, -3.8), (3, y1), and (5, -1). What is
the value of y1 ?
Solution: First, find the slope of the line using the two points (- 2,- 3.8) and
(5, -1).
Find the equation of the line using the slope and either one of the given
points.
Now we can use this equation to find the value of y1 .
Problems:
32. The following three points lie on the same line: ( -4, 4), (2, y1), and
(6,1). What is the value of 1 y1 ?
33. The following three points lie on the same line: ( -5,1.5), ( x1 , -2.7),
and (1, -11.1). What is the
value of x1?
Another way to think of the slope of a line is the average rate of change.
The average rate of change
over an interval is the change in y across the interval divided by the change in
x across the sameinterval.
Example: The table below could be a mathematical
model for some situation. What is the average
rate of change over the interval from 6 to 12?
| x |
0 |
3 |
6 |
9 |
12 |
15 |
18 |
| y |
19 |
17.75 |
17 |
15.5 |
13.5 |
10 |
0 |
Problems: Use the table above to answer the
following problems.
34. What is the average rate of change over the interval from 0 to 3?
35. What is the average rate of change over the interval from 6 to 9?
36. What is the average rate of change over the interval from 3 to 15?
