MA 1128: Lecture 07

1
MA 1128 Lecture 07 2/15/11

• Graphing Linear
• Equations and Functions

2
Graphing Linear Equations

• An equation in two variables, say x and y, is
  linear if it has only x-terms (e.g., 3x),
  y-terms (e.g., -7y), and constant terms (e.g.,
  -2).
• Remember that the terms of an expression are
  separated by additions (subtraction is adding the
  negative).
• For example, 3x y 2y 7 is a linear
  equation,
• since it has an x- and y-term on the left, a
  y-term and constant term on the right, and no
  other kinds of terms.
• The equation 3x2 7y 2 is not a linear
  equation,
• since 3x2 is an x2-term.
• Graphing linear equations (in two variables) is
  easy,
• because the graph is always a straight line.
• We can draw the graph once we have the locations
  of two points on the line.

Next Slide
3
Example

• Consider the linear equation 2x 4y 8.
• We need two solutions.
• We can pick any two values for x, and then find
  the corresponding y-values.
• Lets pick x 0 and x 2.
• If x 0, then 2(0) 4y 8.
• Then 4y 8, and y 2.
• So one solution is (0,2).
• If x 2, then 2(2) 4y 8.
• Then 4 4y 8, 4y 4, and y 1.
• So another solution is (2,1).

Next Slide
4
Example (cont.)

• We can plot these two points (0,2) and (2,1).
  (see the graph below).
• There is only one line that passes through both
  points.
• This is the graph of the equation (see the second
  graph below).

Next Slide
5
Practice Problems

• For the equation 3x 4y 16.
• Find the y-values that correspond to x 0 and
  x 4.
• Plot these points, and draw the graph.
• Click for answers
• 1) y 4 and y 1. 2) Plot the points (0,4)
  and (4,1).

Next Slide
6
One Graphing Strategy

• Pretty much, any two points we use will work.
• Of course, we would like to find two easy points.
• One strategy is to find the point with
  x-coordinate 0, and the point with y-coordinate
  0.
• For example, consider the equation 3x 2y 12.
• If x 0, then 3(0) 2y 12.
• Then 2y 12 and y 6.
• So one solution is (0,6).
• If y 0, then 3x 2(0) 12.
• Then 3x 12, and x 4.
• So another solution is (4,0).
• Note that both of these were very easy to find.

Next Slide
7
Example (cont.)

• The two points and the graph are shown below.
• Note that (0,6) is on the y-axis. This is
  where the line crosses the y-axis, so well say
  that the y-intercept is 6.
• The point (4,0) is on the x-axis, so the
  x-intercept is 4.

Next Slide
8
Example

• Consider the equation 5x 2y 10.
• You can make a little table to keep track of the
  intercepts.
• For the first one, x is zero, so 5x is zero, and
  I just hold my hand over the x-term.
• What times ?2 is 10? y ?5. Keep going.

x y
0
0
x y
0 -5
2 0
Next Slide
9
Practice Problems

• Consider the equation ?3x 4y 12.
• Find the x- and y-intercepts.
• Plot the points and graph the line.
• Click for answers
• 1) The x-intercept is ?4 (when y is zero),
  and the y-intercept is 3.
• 2)

Next Slide
10
Special Cases

• If we had an equation like 2y 3x 2y 6,
• we could subtract 2y from both sides to get
• 3x ?6,
• and then x ?2.
• The y has gone away, but this still is an
  equation in two variables.
• What this means, essentially, is that y can be
  anything.
• Therefore, points like (-2,0), (-2,-7), (-2,5),
  and (-2,-3/2) are all solutions.
• All we need for these to satisfy the equations is
  for the x to be ?2.

Next Slide
11
Example (cont.)

• The four points, (-2,0), (-2,-7), (-2,5), and
  (-2,-3/2), are plotted below.
• Clearly, the line that contains them is vertical.

Next Slide
12
Special Cases (cont.)

• If were talking about lines, or we just know
  that there are supposed to be two variables,
• then linear equations like x -2, x 5, and
  y 3 are just vertical or horizontal lines.
• For y 3, since all of the y-coordinates have
  to be the same (y 3 always), the line must be
  horizontal (see the graph below).
• Its probably easiest to just remember these as
  special cases.

Next Slide
13
Practice Problems

• Graph the line x 3.
• Graph the line y ?2.
• Click for answers
• 1)
  2)

Next Slide
14
Slope

• The most important aspect of the graph of a line
  is a quantity called the slope.
• The slope is just a fraction telling us how much
  the y-coordinate changes divided by how much the
  x-coordinate changes.
• Well think of upward changes in y as positive,
  and downward changes as negative.
• Well also think of changes to the right in x
  as positive, and changes to the left as negative.
  
• With this in mind, the slope is (well always use
  m for slope)

Next Slide
15
Example

• In the picture below, if we start at a point on
  the line and move down 3 and right 2, we get back
  to the line.
• Change in y is 3, and the change in x is 2.
  The slope is m -3/2.
• We could also go left 4 and up 6. The change in y
  is 6,
• and the change in x is 4. The slope is m
  6/(-4) -3/2.
• It doesnt matter which two points we use, after
  simplifying,
• we get the fraction m -3/2.

Next Slide
16
Practice Problems

• Note that (-5,-2) and (5,4) are points on this
  line. The x-int is about 1.67. The spacing of
  the tick marks on the x-axis are 0.4, and theyre
  0.2 on the y-axis.
• Look at the line in the graph below. Start at any
  point you want, and go to the right 5. How far up
  do you have to go to get back to the line?
• If you went down 6, how far (left or right) do
  you have to go, and is it to the right or left?
• What is the slope of this line? Click for
  answers.

1) Up 3 2) Left 10 3) m 3/5
Next Slide
17
More on Slopes

• We can find the slope of a line pretty easily by
  looking at the coordinates of two points on the
  line.
• For example, if (2,5) and (-1,1) are on the
  line,
• and we want to go from (2,5) to (-1,1), the y
  goes from 5 down to 1, so y changes ?4.
  (1) (5) ?4.
• The x goes from 2 to the left to -1, so the
  x changes 3. (-1) (2) -3
• The slope is m (change in y)/(change in x)
  (-4)/(-3) 4/3.
• We could do this just as easily the other way.
• Change in y 1 up to 5 (5) (1) 4
• Change in x -1 right to 2 (2) (-1) 3
• Slope is m 4/3.

Next Slide
18
Practice Problems

• Find the slope if (3,7) and (4,5) are on the
  line.
• Find the slope if (-2,3) and (3,-1) are on the
  line.
• Click for answers
• 1) m ?2 2) m ?4/5.

Next Slide
19
Linear Functions/Slope-Intercept Form

• Consider the linear equation y 2x 6.
• We can find two solutions (0,?) and (2, ?), by
  substituting x 0 and x 2 into the equation.
• We would get (0,-6) and (2, -2).
• The slope is m (-2) (-6)/(2) (0) 4/2
  2.
• Note that (0,-6) is the y-intercept.
• The slope is m 2, and the x-term is 2x.
• The y-intercept is b -6, and the constant
  term is -6.
• It is always true that when a linear equation is
  written so that y is a function of x,
• the coefficient of the x-term is the slope, and
  the constant term is the y-intercept.
• Equations in this form are said to be in
  slope-intercept form.
• y mx b

Next Slide
20
Linear Functions

• In function notation, we replace the y with
  f(x),
• f(x) mx b,
• and any function of this form is called a linear
  function.
• Remember that in f(x) mx b, m is the slope,
  and b is the y-intercept.
• Example. For the linear function f(x) ?7x
  3,
• The slope is m ?7, and the y-intercept is b
  3.

Next Slide
21
Practice Problems

• For the function f(x) (3/2)x 4, what is the
  slope and the y-intercept?
• For the function f(x) ?4x 1, what is the
  slope and the y-intercept?
• Click for answers.
• 1) m 3/2 and b ?4
• 2) m ?4 and b 1.

Next Slide
22
Graphing Linear Functions

• Its really easy to graph a linear function,
  since we know the slope and y-intercept.
• For example, the equation y (3/5)x 2 has m
  3/5 and b ?2.
• One point on the line, therefore, must be
  (0,-2).
• The slope is m 3/5 (3)/(5) (up 3)/(rt 5),
  so we can find a second point.
• Just go right 5 and up 3 (or up 3 and right 5)

Next Slide
23
Example

• Consider the linear function f(x) (-4/3)x 2.
• The y-intercept is b 2.
• From there go down 4 and right 3 to get the
  other point.
• We could also go up 4 and left 3, since
  (-4)/(3) (4)/(-3).

Next Slide
24
Practice Problems

• Let y 3x 2.
• What are the coordinates of the y-intercept?
• Since m 3 3/1 (up 3)/(rt 1), what second
  point does this give you?
• Graph the line.
• Click for answers.
• 1) (0,-2)
• 2) (1,1)
• 3)

End