Intermediate Algebra: A Graphing Approach

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Intermediate Algebra A Graphing Approach

• 2.1 Introduction to Graphing
• 2.2 Introduction to Functions
• 2.3 Graphing Linear Functions
• 2.4 The Slope of a Line

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Section 2.1

• Ordered pair a sequence of 2 numbers where the
  order of the numbers is important
• Axis horizontal or vertical number line
• Origin point of intersection of two axes
• Quadrants regions created by intersection of 2
  axes
• Location of a point residing in the rectangular
  coordinate system created by a horizontal (x-)
  axis and vertical (y-) axis can be described by
  an ordered pair. Each number in the ordered pair
  is referred to as a coordinate.

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Note that the order of the coordinates is very
important, since (-4, 2) and (2, -4) are located
in different positions.
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• Recall that the rectangular coordinate system
  extends infinitely in all directions with the
  axes that form it.
• A graphing utility can only be used to display
  a portion of this system.
• The portion being viewed is called a viewing
  window or simply a window.
• In order to set the window for viewing, you
  have to input the smallest value that you want to
  show on the x-axis (Xmin), the largest value on
  the x-axis (Xmax), and the number of units that
  you want each tick mark on the x-axis to
  represent (Xscl). You have to enter similar
  information for the y-axis, as well (Ymin, Ymax,
  Yscl).
• A standard window going from -10 to 10 with a
  scale of 1 on each axis is generally set as a
  default, but you have to practice changing the
  window settings in the event the standard window
  does not show the region you are interested in.

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• Many graphing utilities will allow you to
  access a particular location on the viewing
  window and move around the viewing window by
  means of a cursor.
• The location of the cursor is designated by an
  ordered pair and you can move the cursor with the
  arrow keys.
• Additional windows, such as an integer window
  or a decimal window give different results for a
  viewing window with some graphing utilities.
• It is important that you read your manual to
  see what window options are available to you, and
  how to set values to produce each of the windows,
  as appropriate.

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• An order pair is a solution of an equation in two
  variables if replacing the variables by the
  appropriate values of the ordered pair results in
  a true statement.

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Determine whether (3, -2) is a solution of 2x
5y -4. Let x 3 and y -2 in the equation.
2x 5y -4 2(3) 5(-2) -4
(replace x with 3 and y with 2) 6 -10
-4 (compute the products)
-4 -4 (True)
So (3, -2) is a solution of 2x 5y -4.
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Determine whether (-1, 6) is a solution of 3x - y
5. Let x -1 and y 6 in the equation.
3x - y 5 3(-1) - 6 5 (replace x
with -1 and y with 6) -3 - 6 5
(compute the product) -9 5
(False)
So (-1, 6) is not a solution of 3x - y 5.
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• Linear equation in two variables
• Ax By C
• A, B, C are real numbers, A B not both 0.
• This is called standard form.
• Graphing linear equations
• Find at least 2 points on the line.
• Connect the points to form a line.

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Graph the linear equation 2x - y -4.
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Graph the linear equation 2x - y -4. Let x 1.
Then 2x - y -4 becomes
2(1) y -4 (replace x with 1) 2
y -4 (simplify the left side)
-y -4 2 -6 (subtract 2 from both
sides) y 6 (multiply both
sides by -1) So one solution is (1, 6).
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Graph the linear equation 2x - y -4. For the
second solution, let y 4. Then 2x - y -4
becomes
2x 4 -4 (replace y with 4)
2x -4 4 (add 4 to both sides)
2x 0 (simplify the right side)
x 0 (divide both sides by
2) So the second solution is (0, 4).
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Graph the linear equation 2x - y -4. For the
third solution, let x -3. Then 2x - y -4
becomes
2(-3) y -4 (replace x with -3)
-6 y -4 (simplify the left side)
-y -4 6 2 (add 6 to both
sides) y -2 (multiply
both sides by -1) So the third solution is (-3,
-2).
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Now we plot all three of the solutions (1, 6),
(0, 4) and (-3, -2).
And then we draw the line that contains the three
points.
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To use a graphing utility to check that our graph
is correct, we need to use the Y editor.
However, our original equation is not in the
appropriate form. We need to solve the equation
for the variable y. 2x - y -4
-y -2x 4 (subtracting 2x from both
sides) y 2x 4
(multiplying both sides by -1) Now we can define
y1 2x 4 and an appropriate window to verify
that we have graphed the equation correctly.
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Let x 4.
y 3 3 6 (simplify the right
side) So one solution is (4, 6).
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For the second solution, let x 0.
y 0 3 3 (simplify the right
side) So a second solution is (0, 3).
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For the third solution, let x -4.
y -3 3 0 (simplify the right
side) So the third solution is (-4, 0).
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Now we plot all three of the solutions (4, 6),
(0, 3) and (-4, 0).
And then we draw the line that contains the three
points.
And we again should check our answer by using a
graphing utility.
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• Intercepts of axes (where graph crosses the axes)
• Since all points on the x-axis have a
  y-coordinate of 0, to find x-intercept, let y 0
  and solve for x.
• Since all points on the y-axis have an
  x-coordinate of 0, to find y-intercept, let x 0
  and solve for y.

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• Besides linear equations, graphs of 3 other
  types of equations are examined in this section.
• Well examine these types of equations in more
  detail later, but you should know their general
  shape and thus be able to roughly plot the
  equations by finding points that solve it.

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• The first type of non-linear equation is a
  quadratic.
• Has ONE of the variables squared (well use the
  x being squared in this section).
• It is shaped like a narrow, steep cup or
  mountain.
• Plug values in the equation for the x and find
  the corresponding y values to get your points.
• Might need a LOT of points go until you find
  where the graph starts curving back.

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Graph y 2x2 4.
2
4
1
-2
0
-4
-1
-2
-2
4
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• The second type of non-linear equation is a
  cubic.
• There are four general versions of what a cubic
  equation in the variable x might look like.
• The form of the cubic equation will be y ax3
  bx2 cx d, where a, b, c, d are real
  numbers, but a ? 0.
• Using a graphing utility, try various
  combinations of real numbers in place of the
  coefficients of the powers of x in the equation
  to see if you can determine the 4 basic shapes.

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• The last type of non-linear equation in this
  section involves absolute values.
• As with quadratics and cubics, you need to know
  the general shape (we will study in more detail
  later), so that you can plot points and sketch it
  in.
• It will look like a V opening up or an upside
  down V.

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Graph y x - 3.
2
-1
1
-2
0
-3
-1
-2
-1
-2
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Section 2.2

• Equations in two variables define relations
  between the two variables.
• There are other ways to describe relations
  between variables.
• Set to set
• Ordered pairs

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• A set of ordered pairs is also called a relation.
• The domain is the set of x-coordinates of the
  ordered pairs.
• The range is the set of y-coordinates of the
  ordered pairs.

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• Find the domain and range of the relation (4,9),
  (-4,9), (2,3), (10,-5).
• Domain is the set of all x-values, 4, -4, 2,
  10.
• Range is the set of all y-values, 9, 3, -5.

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Find the domain and range of the following
relation.

• Input (Animal)
• Polar Bear
• Cow
• Chimpanzee
• Giraffe
• Gorilla
• Kangaroo
• Red Fox

• Output (Life Span)
• 20
• 15
• 10
• 7

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Domain is Polar Bear, Cow, Chimpanzee, Giraffe,
Gorilla, Kangaroo, Red Fox Range is 20, 15, 10,
7
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• Some relations are also functions.
• A function is a set of order pairs in which each
  first component in the ordered pairs corresponds
  to exactly one second component.

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• Given the relation (4,9), (-4,9), (2,3),
  (10,-5), is it a function?
• Since each element of the domain is paired with
  only one element of the range, it is a function.
• Note Its okay for a y-value to be assigned to
  more than one x-value, but an x-value cannot be
  assigned to more than one y-value (has to be
  assigned to ONLY one y-value).

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• Is the relation y x2 2x a function?
• Since each element of the domain (the x-values)
  would produce only one element of the range (the
  y-values), it is a function.

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• Is the relation x2 y2 9 a function?
• Since each element of the domain (the x-values)
  would correspond with 2 different values of the
  range (both a positive and negative y-value), the
  relation is NOT a function.

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• Relations and functions can also be described by
  graphing their ordered pairs.
• Graphs can be used to determine if a relation is
  a function.
• If an x-coordinate is paired with more than one
  y-coordinate, a vertical line can be drawn that
  will intersect the graph at more than one point.
• If no vertical line can be drawn so that it
  intersects a graph more than once, the graph is
  the graph of a function. This is the vertical
  line test.

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Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since no vertical line will intersect this graph
more than once, it is the graph of a function.
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Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since no vertical line will intersect this graph
more than once, it is the graph of a function.
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Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since vertical lines can be drawn that intersect
the graph in two points, it is NOT the graph of a
function.
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Use the vertical line test to determine whether
the graph to the right is the graph of a function.
Since a vertical line can be drawn that
intersects the graph at every point, it is NOT
the graph of a function.
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• Since the graph of a linear equation is a line,
  all linear equations are functions, except those
  whose graph is a vertical line.

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Find the domain and range of the function graphed
to the right. Use interval notation.
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Find the domain and range of the function graphed
to the right. Use interval notation.
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• Specialized notation is often used when we know a
  relation is a function and it has been solved for
  y.
• For example, the graph of the linear equation
  y -3x 2 passes the vertical line test, so it
  represents a function.
• We often use letters such as f, g, and h to name
  functions.
• We can use the function notation f(x) (read f of
  x) and write the equation as f(x) -3x 2.
• Note The symbol f(x) is a specialized notation
  that does NOT mean f x (f times x).

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• When we want to evaluate a function at a
  particular value of x, we substitute the x-value
  into the notation.
• For example, f(2) means to evaluate the function
  f when x 2. So we replace x with 2 in the
  equation.
• For our previous example when f(x) -3x 2,
  f(2) -3(2) 2 -6 2 -4.
• When x 2, then f(x) -4, giving us the ordered
  pair (2, -4).

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• Given that g(x) x2 2x, find g(-3). Then
  write down the corresponding ordered pair.
• g(-3) (-3)2 2(-3) 9 (-6) 15.
• The ordered pair is (-3, 15).

To evaluate g(-3) with a graphing utility, we
could substitute -3 into the function, as above,
or we could use the store feature. We store -3
into x, then find x2 2x with this value.
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Given the graph of the following function, find
each function value by inspecting the graph.
7
f(5)
3
f(4)
-1
f(-5)
-7
f(-6)
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• You can use a graphing utility to evaluate a
  function by using a graph, as well.
• Once you use the Y editor to graph the
  function, you can move the cursor to the point on
  the graph with the particular x-coordinate you
  are interested in.
• Then look at the y-coordinate of the point to
  determine your evaluated function.

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Section 2.3
Linear function is a function that can be written
in the form f(x) mx b.
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Graph the following functions on the same pair of
axes.

• f(x) -2x

• g(x) -2x 3

• h(x) -2x - 5

Note You would use the Y editor and y1, y2, y3
on a graphing utility.
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If you use a graphing utility to graph several
families of linear functions (as in the last
example), you would see that the graph of y
f(x) K is the same as y f(x) shifted K
units upward if K is positive and downward if K
is negative.
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• Where do each of the graphs intersect the
  y-axis in the previous example?
• At (0,0), (0,3) and (0,-5) respectively.
• These are referred to as the y-intercepts.
• When a linear function is written in the form
  of f(x) mx b or y mx b, the
  y-intercept is (0, b).

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• How would we find the y-intercept if the line is
  not written in the form f(x) mx b or y mx
  b?
• Since all points on the y-axis have an
  x-coordinate of 0, substitute x 0 into the
  linear equation and find the corresponding
  y-coordinate.

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• Find the y-intercept of 4 x 3y.
• Let x 0.
• Then 4 x 3y becomes
• 4 0 3y (replace x with 0)
• 4 -3y (simplify the right side)

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• Correspondingly, all points on the x-axis would
  have a y-coordinate of 0.
• So to find an x-intercept of a linear function,
  we would substitute y 0 into the equation and
  find the corresponding x-coordinate.

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• Find the x-intercept of 4 x 3y.
• Let y 0.
• Then 4 x 3y becomes
• 4 x - 3(0) (replace y with 0)
• 4 x (simplify the right side)
• So the x-intercept is (4,0).

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• Graph the linear equation 4 x 3y by plotting
  intercepts.

Plot both of these points and then draw the line
through the 2 points. Note You should still
find a 3rd solution to check your computations.
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Graph the linear equation 4 x 3y. Along with
the intercepts, for the third solution, let y
1. Then 4 x 3y becomes
4 x 3(1) (replace y with 1)
4 x 3 (simplify the right side) 4
3 x (add 3 to both sides) 7 x
(simplify the left side) So the third
solution is (7, 1).
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And then we draw the line that contains the three
points.
To check your work with a graphing utility, you
would have to solve the equation for y and use
the Y editor.
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• Graph 2x y by plotting intercepts.
• To find the y-intercept, let x 0.
• 2(0) y
• 0 y, so the y-intercept is
  (0,0).
• To find the x-intercept, let y 0.
• 2x 0
• x 0, so the x-intercept is
  (0,0).
• Oops! Its the same point. What do we do?

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• Since we need at least 2 points to graph a line,
  we will have to find at least one more point.
• Let x 3
• 2(3) y
• 6 y, so another point is (3, 6).
• Let y 4
• 2x 4
• x 2, so another point is (2, 4).

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Now we plot all three of the solutions (0, 0),
(3, 6) and (2, 4).
And then we draw the line that contains the three
points.
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• Graph y 3.
• Note that this line can be written as y 0x
  3.
• The y-intercept is (0, 3), but there is no
  x-intercept!
• (Since an x-intercept would be found by letting
  y 0, and 0 ? 0x 3, there is no x-intercept.)
• Every value we substitute for x gives a
  y-coordinate of 3.
• The graph will be a horizontal line through the
  point (0,3) on the y-axis.

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• Graph x -3.
• This equation can be written x 0y 3.
• When y 0, x -3, so the x-intercept is (-3,0),
  but there is no y-intercept.
• Any value we substitute for y gives an
  x-coordinate of 3.
• So the graph will be a vertical line through the
  point (-3,0) on the x-axis.

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• Vertical lines
• Appear in the form of x c, where c is a real
  number.
• x-intercept is at (c, 0), no y-intercept unless
  c 0 (y-axis).
• Horizontal lines
• Appear in the form of y c, where c is a real
  number.
• y-intercept is at (0, c), no x-intercept unless
  c 0 (x-axis).

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Section 2.4

• Slope of a line
• Informally, slope is the tilt of a line.
• It is the ratio of vertical change to horizontal
  change, or

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• Find the slope of the line through (4, -3) and
  (2, 2).
• If we let (x1, y1) be (4, -3) and (x2, y2) be (2,
  2), then

Note If we let (x1, y1) be (2, 2) and (x2, y2)
be (4, -3), then we get the same result.
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• Find 2 points on the graph, then use those points
  in the slope formula.

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• Slope-intercept form of a line
• y mx b has a slope of m and has a
  y-intercept of (0, b).
• This form is useful for graphing, since you have
  a point and the slope readily visible from the
  equation.

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• Find the slope and y-intercept of the line 3x
  y -5.
• First, we need to solve the linear equation for
  y.
• By adding 3x to both sides, y 3x 5.
• Once we have the equation in the form of y mx
  b, we can read the slope and y-intercept.
• slope is 3
• y-intercept is (0,-5)

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• Find the slope and y-intercept of the line 2x
  6y 12.
• First, we need to solve the linear equation for
  y.
• -6y -2x 12 (subtract 2x from both sides)

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• For any 2 points, the y values will be equal to
  the same real number.
• The numerator in the slope formula 0 (the
  difference of the y-coordinates), but the
  denominator ? 0 (two different points would have
  two different x-coordinates).
• So the slope 0.

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• For any 2 points, the x values will be equal to
  the same real number.
• The denominator in the slope formula 0 (the
  difference of the x-coordinates), but the
  numerator ? 0 (two different points would have
  two different y-coordinates),
• So the slope is undefined (since you cant divide
  by 0).

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• If a line moves up as it moves from left to
  right, the slope is positive.
• If a line moves down as it moves from left to
  right, the slope is negative.
• Horizontal lines have a slope of 0.
• Vertical lines have undefined slope (or no slope).

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• Two lines that never intersect are called
  parallel lines.
• Parallel lines have the same slope
• unless they are vertical lines, which have no
  slope.
• Vertical lines are also parallel.

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• Two lines that intersect at right angles are
  called perpendicular lines.
• Two nonvertical perpendicular lines have slopes
  that are negative reciprocals of each other.
• The product of their slopes will be 1.
• Horizontal and vertical lines are perpendicular
  to each other.

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• Determine whether the following lines are
  parallel, perpendicular, or neither.
• -5x y -6 and x 5y 5
• First, we need to solve both equations for y.
• In the first equation,
• y 5x 6 (add 5x to both sides)
• In the second equation,
• 5y -x 5 (subtract x from both sides)

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• Determine whether the following lines are
  parallel, perpendicular, or neither.
• -x 2y -2 and 2x 4y 3
• In the first equation,
• 2y x 2 (add x to both sides)

• In the second equation,
• 4y 2x 3 (subtract 3 from both
  sides)