Graphing Equations and Inequalities

1
Graphing Equations and Inequalities
Chapter 10
2
Chapter Sections
10.1 Reading Graphs and the Rectangular
Coordinate System 10.2 Graphing Linear
Equations 10.3 Intercepts 10.4 Slope and
Rate of Change 10.5 Equations of Lines 10.6
Introduction to Functions 10.7 Graphing Linear
Inequalities in Two Variables 10.8 Direct and
Inverse Variation
3
10.1

• Reading Graphs and the Rectangular Coordinate
  System

4
Vocabulary

• 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

5
Graphing an Ordered Pair
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Graphing an Ordered Pair
Note that the order of the coordinates is very
important, since (-4, 2) and (2, -4) are located
in different positions.
7
Vocabulary

• Paired data are data that can be represented as
  an ordered pair
• A scatter diagram is the graph of paired data as
  points in the rectangular coordinate system
• 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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Solutions of an Equation
Example
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
9
Solutions of an Equation
Example
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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Solving an Equation
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Solving an Equation
Example
Complete the ordered pair (4, ) so that it is a
solution of 2x 4y 4.
Let x 4 in the equation and solve for y.
2x 4y 4 2(4) 4y 4
(replace x with 4) 8 4y 4
(compute the product)
So the completed ordered pair is (4, 3).
4y 12 (simplify both sides)
y 3 (divide both sides by 4)
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Solving an Equation
Example
Complete the ordered pair (__, 2) so that it is
a solution of 4x y 4.
Let y 2 in the equation and solve for x.
4x y 4 4x ( 2) 4
(replace y with 2) 4x 2 4
(simplify left side)
So the completed ordered pair is (½, 2).
4x 2 (simplify both sides)
x ½ (divide both sides by 4)
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10.2

• Graphing Linear Equations

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Linear Equations

• Linear Equation in Two Variables
• Ax By C
• A, B, C are real numbers, A and B not both 0
• This is called standard form
• Graphing Linear Equations
• Find at least 2 points on the line
• y mx b crosses the y-axis at b (called
  slope-intercept form)

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Graphing Linear Equations
Example
Graph the linear equation 2x y 4.
Continued.
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Graphing Linear Equations
Example continued
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)
Continued.
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Graphing Linear Equations
Example continued
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)
Continued.
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Graphing Linear Equations
Example continued
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)
Continued.
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Graphing Linear Equations
Example continued
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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Graphing Linear Equations
Example
Continued.
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Graphing Linear Equations
Example continued
Let x 4.
y 3 3 6 (simplify the right
side) So one solution is (4, 6)
Continued.
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Graphing Linear Equations
Example continued
For the second solution, let x 0.
y 0 3 3 (simplify the right
side) So a second solution is (0, 3)
Continued.
23
Graphing Linear Equations
Example continued
For the third solution, let x 4.
y 3 3 0 (simplify the
right side) So the third solution is ( 4, 0)
Continued.
24
Graphing Linear Equations
Example continued
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.
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10.3

• Intercepts

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Intercepts

• 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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Intercepts
Example

• 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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Intercepts
Example

• 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 by Plotting Intercepts
Example

• 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.
Continued.
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Graph by Plotting Intercepts
Example continued
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)
Continued.
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Graph by Plotting Intercepts
Example continued
And then we draw the line that contains the three
points.
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Graph by Plotting Intercepts
Example

• 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?

Continued.
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Graph by Plotting Intercepts
Example continued

• 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)

Continued.
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Graph by Plotting Intercepts
Example continued
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 by Plotting Intercepts
Example

• 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

Continued.
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Graph by Plotting Intercepts
Example continued
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Graph by Plotting Intercepts
Example

• 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

Continued.
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Graph by Plotting Intercepts
Example continued
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Vertical and Horizontal Lines

• 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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10.4

• Slope and Rate of Change

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Slope

• Slope of a Line

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Slope
Example

• 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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Slope of a Horizontal Line

• 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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Slope of a Vertical Line

• 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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Summary of Slope of Lines

• 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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Parallel Lines

• 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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Parallel Lines
Example

• Find the slope of a line parallel to the line
  passing through (0,3) and (6,0)

So the slope of any parallel line is also ½
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Perpendicular Lines

• 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

49
Perpendicular Lines
Example

• Find the slope of a line perpendicular to the
  line passing through (?1,3) and (2,-8)

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Parallel and Perpendicular Lines
Example

• 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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10.5

• Equations of Lines

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Slope-Intercept Form

• 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.

53
Slope-Intercept Form
Example

• 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)

54
Slope-Intercept Form
Example

• 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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Point-Slope Form

• The slope-intercept form uses, specifically, the
  y-intercept in the equation.
• The point-slope form allows you to use ANY point,
  together with the slope, to form the equation of
  the line.

m is the slope (x1, y1) is a point on the line
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Point-Slope Form
Example

• Find an equation of a line with slope 2, through
  the point (11, 12). Write the equation in
  standard form.
• First we substitute the slope and point into the
  point-slope form of an equation.
• y ( 12) 2(x ( 11))
• y 12 2x 22 (use distributive
  property)
• 2x y 12 22 (add 2x to both sides)
• 2x y 34 (subtract 12 from both
  sides)

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Point-Slope Form
Example

• Find the equation of the line through (4,0) and
  (6, 1). Write the equation in standard form.
• First find the slope.

Continued.
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Point-Slope Form
Example continued
Now substitute the slope and one of the points
into the point-slope form of an equation.
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Point-Slope Form
Example
Find the equation of the line passing through
points (2, 5) and (4, 3). Write the equation
using function notation.
Continued.
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Point-Slope Form
Example continued
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10.6

• Introduction to Functions

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Vocabulary

• An equation in 2 variables defines a relation
  between the two variables.
• 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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Domain and Range
Example

• 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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Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
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Domain and Range
Example
Find the domain and range of the function graphed
to the right. Use interval notation.
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Functions

• Some relations are also functions.
• A function is a set of order pairs that assigns
  to each x-value exactly one y-value.

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Functions
Example

• 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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Vertical Line Test

• 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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Vertical Line Test
Example
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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Vertical Line Test
Example
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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Vertical Line Test
Example
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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Vertical Line Test

• 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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Function Notation

• 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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Function Notation

• 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 order
  pair (2, 4).

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Function Notation
Example

• 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).

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10.7

• Graphing Linear Inequalities in Two Variables

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Linear Equations in Two Variables

• Linear inequality in two variables
• Written in the form Ax By lt C
• A, B, C are real numbers, A and B are not both 0
• Could use (gt, ?, ?) in place of lt
• An ordered pair is a solution of the linear
  inequality if it makes the inequality a true
  statement.

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Linear Equations in Two Variables

• To Graph a Linear Inequality
• Graph the related linear equality (forms the
  boundary line).
• ? and ? are graphed as solid lines
• lt and gt are graphed as dashed lines
• Choose a point not on the boundary line
  substitute into original inequality.
• If a true statement results, shade the half-plane
  containing the point.
• If a false statements results, shade the
  half-plane that does NOT contain the point.

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Linear Equations in Two Variables
Example

• Graph 7x y gt 14

• Graph 7x y 14 as a dashed line.

• Pick a point not on the graph
  (0,0)

• Test it in the original inequality.
• 7(0) 0 gt 14, 0 gt 14
• True, so shade the side containing (0,0).

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Linear Equations in Two Variables
Example

• Graph 3x 5y ? 2

• Graph 3x 5y 2 as a solid line.

• Pick a point not on the graph
  (0,0), but just barely

• Test it in the original inequality.
• 3(0) 5(0) gt 2, 0 gt 2
• False, so shade the side that does not contain
  (0,0).

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Linear Equations in Two Variables
Example

• Graph 3x lt 15

• Graph 3x 15 as a dashed line.

• Pick a point not on the graph
  (0,0)

• Test it in the original inequality.
• 3(0) lt 15, 0 lt 15
• True, so shade the side containing (0,0).

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Linear Equations in Two Variables
Warning!

• Note that although all of our examples allowed us
  to select (0, 0) as our test point, that will not
  always be true.
• If the boundary line contains (0,0), you must
  select another point that is not contained on the
  line as your test point.

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10.8

• Direct and Inverse Variation

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Direct Variation

• y varies directly as x, or y is directly
  proportional to x, if there is a nonzero constant
  k such that y kx.
• The family of equations of the form y kx are
  referred to as direct variation equations.
• The number k is called the constant of variation
  or the constant of proportionality.

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Direct Variation

• If y varies directly as x, find the constant of
  variation k and the direct variation equation,
  given that y 5 when x 30.
• y kx
• 5 k30
• k 1/6

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Direct Variation
Example

• If y varies directly as x, and y 48 when x 6,
  then find y when x 15.
• y kx
• 48 k6
• 8 k
• So the equation is y 8x.
• y 815
• y 120

87
Direct Variation
Example

• At sea, the distance to the horizon is directly
  proportional to the square root of the elevation
  of the observer. If a person who is 36 feet
  above water can see 7.4 miles, find how far a
  person 64 feet above the water can see. Round
  your answer to two decimal places.

Continued.
88
Direct Variation
Example continued
We substitute our given value for the elevation
into the equation.
So our equation is
89
Inverse Variation

• y varies inversely as x, or y is inversely
  proportional to x, if there is a nonzero constant
  k such that y k/x.
• The family of equations of the form y k/x are
  referred to as inverse variation equations.
• The number k is still called the constant of
  variation or the constant of proportionality.

90
Inverse Variation
Example

• If y varies inversely as x, find the constant of
  variation k and the inverse variation equation,
  given that y 63 when x 3.
• y k/x
• 63 k/3
• k 633
• k 189

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Powers of x

• y can vary directly or inversely as powers of x,
  as well.
• y varies directly as a power of x if there is a
  nonzero constant k and a natural number n such
  that y kxn

92
Powers of x
Example

• The maximum weight that a circular column can
  hold is inversely proportional to the square of
  its height.
• If an 8-foot column can hold 2 tons, find how
  much weight a 10-foot column can hold.

Continued.
93
Powers of x
Example continued
We substitute our given value for the height of
the column into the equation.
So our equation is