Graphing in the Coordinate Plane

1
Graphing in the Coordinate Plane

• Chapter 3

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Aim 3-1 How do we identify and graph points on
the coordinate plane?

• Mapmakers use a grid system with rectangular
  sections to help people read a map.
• Turn to page 117 Complete the check for
  understanding prob. 1

3
Review Key Terms

• Coordinate plane is the grid that is divided into
  four quadrants by the x and y-axis.
• X-axis is the horizontal number line.
• Y-axis the vertical number line.
• (2, -4) is an example of an ordered pair.
• The first point is the x and the second is the y.

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SummaryAnswer in a complete sentence.

• Explain how an ordered pair locates a point in
  the coordinate plane.

5
Warm-up

• Evaluate each expression for a 4.
• 1.6a 21
• 2.13 2a
• 3. 0.5a 8
• 4. 2.4a 7
• 5. 33 - 5a
• 6. 1.6 (9 - a)

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Aim 3-2 How do we solve equations with 2
variables?

• Investigation Finding Solutions
• Suppose your soccer coach e-mails your team about
  an optional practice. There are 12 people on your
  team.
• If a given number of people can make the
  practice, how many cannot? Copy, extend and
  complete the table below to show the possible
  answers.

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Investigation continued

• 2. You can write the answers in the table as
  ordered pairs (x, y).
• Let x the number of people who can attend
  practice.
• Let y the number of people who cannot attend
  practice.
• The ordered pair (9,3) means 9 can attend and 3
  cannot. Write all the answers in a table as an
  ordered pair.
• 3.Graph each ordered pair.
• Describe the pattern of the points.

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• The goal of this lesson is to solve and graph
  equations with two variables.
• The equation we can write for the investigation
  scenario is x y 12.
• Any ordered pair that makes the equation true is
  a solution.
• Example 3 9 12

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Finding Solutions

• Determine whether (-3,2) is a solution of
• y 2x 8.
• Substitute the given coordinates into the
  equation.
• 2 2(-3) 8
• 2 - 6 8
• 2 2 ? It is a solution.

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Practice

• Determine whether each ordered pair is a solution
  of y 3x 4.
• a) (5, 10) b) (-2, - 10)

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Real-World Problem Solving

• Suppose you saved 38. In the equation
• y 38 4x, y is the amount you have saved after
  x weeks if you save 4 each week.
• Complete the solution (12, _ ) to find how much
  money you have saved after 12 weeks.

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Practice

• Use the equation y 3x 2. Complete each
  solution.
• a. (2, ?) b. ( 5, ?)

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How do we graph equations with two variables?

• An equation with two variables can have many
  solutions. The graph of a linear equation like y
  38 4x lie on a line.

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• Graph y -1/2 x 2
• Step 1 Make a table.
• Pick at least 5 x points for your table.
• Step 2 Graph the ordered pairs and draw line
  through the points. Use a ruler.

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Practice

• Graph the linear equation.
• y - 3x 2

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Summary Answer in a complete sentence.

• How do you graph a linear equation with two
  variables?

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Warm-up

• Simplify each expression.
• -3 1
• 4 2
• 10 (-4)
• 1 7
• - 8 (-6)
• -5 9

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Aim 3-3 How do we find the slope of a line?

• Slope is a number that tells steepness.
• Slope vertical change rise
• horizontal change run
• Slope change in y
• change in x
• A positive slope means a line is increasing while
  a negative slope means a line is decreasing.

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• Finding the slope of a line.
• (x, y)
• Given the two points A (2, 1)
• B(5, -3)
• Slope 1- (-3) 4
• 2- 5 -3
• Is the line increasing or decreasing? Explain.

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Guided Practice

• Given the two points find the slope.
• (-3, 2) and (2, 4).
• slope 2 4 - 2 2
• -3 2 -5 5
• Remember When you decide which point to start
  with for the change in y, be sure to start with
  the same point for x.

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Practice

• Given the two points find the slope.
• (-2, 2) and (2, 0)

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Special Slopes

• Horizontal line has a slope of 0.
• A vertical line has an undefined slope.

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Find the slopes for the given points. Then
identify what kind of line it is.

• 1. (3, 1) and (3, -2)
• 2. (-5, -3) and (-2, -3)

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Finding the slope from a table.

• Find the change in y.
• 7-5 2
• 5 3 2
• 3- 1 2
• Find the change in x.
• -3 0 -3
• 0 -3 - 3
• 3 - 6 -3
• Now express the slope as a ratio 2/-3
• Since the change was consistent, we can say the
  data is linear.

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SummaryAnswer in a complete sentence.

• Describe how to find the slope of line from a
  graph and from a table.

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Warm-up/ Review

• Create a table for the following linear equation,
  then graph on a new coordinate plane. Use the
  graph paper given last week. Remember, you should
  have at least 5 points.
• y x 5
• Find the slope of each pair of point.
• 1. w(4, 9) U (3, 7)
• 2. X(-2, 10) y ( 0, 7)
• 3. A(-4, -5) B (-1, 0)
• The slope of a vertical line is ____.
• The slope of a horizontal line is ____.

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Aim 3-4 How do we graph an equation in
slope-intercept form?

• An equation written in the form
• y mx b is called in slope-intercept form.
• M stands for the slope and b stands for the
  y-intercept.

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Identifying the slope and y-intercept.

• Example 1
• Y 6x What is slope?
• The slope is 6.
• What is the y-intercept?
• The intercept is 0 or (0,0)
• Example 2
• y -x 4 What is the slope?
• The slope is -1.
• What is the y-intercept?
• The intercept is -4 or (0, -4).

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Identify the slope and y-intercept.

• Y 2x -1
• Y - x
• Y -4/5 x 2

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Graphing a line using the slope and y-intercept.

• Y 3/5 x 4
• First, plot the y-intercept. The y-intercept for
  this equation is (0,4).
• Then from this point using your slope you can
  plot the next point (s).
• Slope 3 rise
• 5 run
• The next point you should have plotted are (5,
  7) and (10, 10).

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• Graph the following line using the slope and
  y-intercept.
• 1. y -3x 4
• 2. Y x
• 3. Y -1/2 x - 3

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Summary Answer in a complete sentence.

• What is a line in slope-intercept form?
• Give an example.
• How do you use the slope and y-intercept to graph
  a line?
• How is using the slope and y-intercept different
  from creating a table? Is it easier?

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Warm-up

• Identify the slope and y-intercept.
• Y x 2 2. y 7x 3. y -2x 8
• Graph the equations using the slope and
  y-intercept.
• 4. Y x 5 5. y -2x 3
• Which of the above lines is increasing and
  decreasing? Could you have known before you
  graphed? Explain.

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Aim 3-4b How do write an equation for a line?

• Given the slope as ½ and y-intercept of -6 you
  can write an equation in the form of
• y mx b.
• Substitute m with the given slope and b with the
  y-intercept.
• Solution y ½ x - 6

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Practice

• Write an equation for a line with the given slope
  and y-intercept.
• m 4, b -2
• m -3/2 b 5
• m ¼ b 0

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Writing an equation for a line using a graph.

• First, identify the y-intercept.
• Where does the line hit the y-axis?
• Then pick points on the line and find the slope.
• Then you can write your equation.

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Summary Answer in a complete sentence.

• Explain how to write an equation of a line using
  a graph.

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Warm-up

• Graph each equation using the slope and
  y-intercept.
• Y x 3 2. y -3x 1
• 3. Y 4x 2 4. y -1/2x 2
• 5. Explain how you would graph the line
• y -1/4x 2.

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Aim 3-5How do we solve a problem by combining
strategies?

• Suppose your family is planning a special party
  at a restaurant for your birthday. A buffet
  dinner costs 15 per person. For dessert, your
  parents plan to buy a birthday cake that costs
  30. Find the number of people you can invite to
  the party for 275.

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• Read and understand Your goal is to estimate the
  number of people you can invite for 275,
  including yourself and your family. The buffet
  costs 15 per person, and the cake costs 30.
  Assume no tax or tip is involved.
• Plan and solve First, write an equation.
• Total cost cost per person of people
  cake
• t total cost p of people
• Equation t 15 p 30

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• Equation t 15 p 30
• Strategy 1
• You can now make a graph for the above equation
  and estimate the number of people you can invite
  for 275.
• Be sure to choose appropriate scales and
  intervals for your axes.
• Find the point on the line with a y-coordinate of
  275. Then estimate your x-coordinate on the
  graph of the line.
• Strategy 2
• Substitute t for 275 and solve for p.
• P 16.33 which means you can invite up to 16
  people.
• Does your estimate agree with calculations?

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Practice

• A server at a local restaurant receives an
  average tip of 2 per customer and a salary of
  10 for a four-hour shift. Write an equation and
  make a graph to represent the servers total
  earnings for a four-hour shift.
• b. Estimation Estimate the total earnings for
  serving 12 customers.

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SummaryAnswer in a complete sentence.

• When is writing a linear equation a useful
  problem solving strategy?
• When is graphing a linear equation a useful
  problem solving strategy?

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Warm-up

• Find the slope.
• (4, 7) and (1, 3) b. (2, 3) and (4, -3)
• You have an equal number of pennies, nickels, and
  quarters. The total is 6.15. How many of each
  type do you have?
• A group of 32 classmates decide to go on a bike
  ride. There are 6 more girls than boys. How many
  boys participate in the bike ride?

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Aim 3-8 How do we graph translations?

• Transformation is a change in the position, shape
  or size of a figure.
• There are 3 types of transformations that change
  the position of the shape but NOT its size. They
  are reflection, rotation and translation.

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• Translation which moves each point in the same
  direction the same number of steps.
• To identify the image of point A after a
  translation is A (A prime).

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• Translating a Point
• Plot the point B(2, -1). Translate point B
• 3 units up and 4 units right.
• What are the coordinates of B'?
• Solution B' (6, 2)

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Translating a Point

• Graph each point and its image after the
  translation.
• a. C (-3, 2) up 4 units and right 7 units.
• What are the coordinates of C'?
• b. D( 4, -3) down 2 units, left 5 units.
• What are the coordinates of D'?

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Number Sense

• Does translating a point up or down affect the
  x-coordinate or the y-coordinate?
• What about translating a point to the left or
  right?

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Translating a Figure

• ?DEF has vertices D(-5, 1), E(-1, 4), and F(-2,
  2). Translate the ?DEF to the right 6 units and
  down 3 units.
• What are the coordinates of ?D'E'F'?
• Solution D'(1, -2) E'(5, 1), F'(4, -1)

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Translating a Figure

• Graph ?GHI, G(3, 3), H(5, -1), I (1, -1).
• Translate ?GHI left 3 units and up 4 units.
• What are the coordinates of ?G'H'I'?

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Describing a Translation

• You can use an arrow notation to write a general
  rule that describes a translation.
• Example right 4, down 2
• (x, y) ? (x 4, y -2)
• left 3, up 3
• (x, y) ? (x -3, y 3
• )

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SummaryAnswer in a complete sentence.

• When a triangle gets translated what stays the
  same and what changes?
• How can one write a general rule to describe a
  translation?

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Warm-up

• A point and its image are given. Write a rule to
  describe the translation.
• S(-3, 8), S'(4, 0)
• Suppose you translate a point to the left 1 unit
  and up 3 units. Describe what you would do to the
  coordinates of the original point to find the
  coordinates of the new image.

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Aim 3-9 How do we graph reflections?

• Reflection is a transformation that flips a
  figure over a line.
• This line is called the line of reflection.
• Like translations reflections change the position
  of the figure but NOT its size or shape.

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Graphing Reflections of a Point

• Graph the point A(3,2). Then graph the point
  after its reflection over the x-axis.
• What are the coordinates of A'?
• Solution A'(3, -2)

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Graphing Reflections of a Point

• Graph the given point and its image after the
  reflection.
• C(-4, 3) over the x-axis.
• What are the coordinates of C'?
• D(-2, 1) over the y-axis.
• What are the coordinates of D'?

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Number Sense

• How do the signs of the coordinates change when
  you reflect a point over the x-axis? Over the
  y-axis?

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Graphing a Reflection of a Shape

• Graph ?BCD B(-4, 4), C (-1, 5), D(-2, 1) and its
  image after it is reflected over the line through
  (1,3) and (1,0).
• Name the coordinates of the vertices of ?B'C'D'.
• Solution B'(6, 4), C'(3, 5), D'(4, 1)

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Graphing a Reflection of a Shape

• ?EFG has vertices E(4, 3), F(3, 1), and G(1, 2).
  Graph ?EFG and its image after a reflection over
  the line (-2, 4) and (-2, 0).
• What are the coordinates of the reflected
  triangle?

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Identifying Lines of Symmetry

• If a figure can be reflected over a line so that
  its image matches the original figure. The figure
  has reflectional symmetry.
• The line the figure reflects over is called the
  line of symmetry.

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SummaryAnswer in a complete sentence.

• Explain how to graph the reflection of a triangle
  over a line. Give an example.
• How many lines of symmetry does a circle have?
  Explain.

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Warm-up

• Chapter 3 Review
• P. 176 prob. 6 13
• Note When graphing, be sure to use the
  y-intercept and slope to graph!

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Aim 3-10 How do we graph rotations?

• A rotation is a transformation that turns a
  figure about a fixed point.
• The fixed point is called the center of rotation.
• Rotation changes the position of the figure but
  NOT its size and shape.
• The angle of rotation is the number of degrees
  the figure rotates.
• In this lesson all rotations are counterclockwise.

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Graphing Rotation

• Graph ?ABC, A(3, 3), B(1, -1), C(3, -2).
• Rotate ?ABC 90 about the origin.
• Trace the vertices of the triangle, the x-axis
  and the y-axis.
• Place your pencil at the origin to rotate the
  paper.
• Rotate tracing paper 90 counterclockwise.
• The axes should line up.
• Mark the position of each vertex by pressing your
  pencil through the paper.
• Complete the new figure. Draw the triangle and
  label the vertices.

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Extension

• Rotate ?ABC (the original) about the origin 180.
• What are new coordinates of ?A'B'C'?

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Identifying Rotational Symmetry

• A figure has rotational symmetry if it can be
  rotated 180 or less and it exactly matches its
  original figure.
• For rotational symmetry its the fewest number of
  degrees the figure must be rotated to match the
  original figure.
• A complete rotation is 360.

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Identifying the Angle of Rotation

• Demonstration

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SummaryAnswer in a complete sentence.

• Describe a rotation.
• Describe a figure with rotational symmetry.

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Warm-up Fill-in the blank with the best word or
phrase.

• The ___ tells the steepness of a line. To find
  the __ you need to find the change in __ over the
  change in __ or the ratio of rise over ___.
• The linear equation y 2x 4 is called ___.
• The 2 stands for the __ and the 4 is the __.

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Chapter Review

• What is an ordered pair?
• What do you call the grid where you graph?
• When graphing which coordinate goes first and
  then second?
• P. 176 complete prob. 6-13

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• Identify the slope and the y-intercept. Then
  graph using the slope and y-intercept.
• Y3x 5
• Additional Practice p. 177 prob. 19, 20

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Transformations

• There are three types that change the position of
  a figure but do NOT change the size and shape.
• They are translation also known as a slide,
• reflection and rotation.
• For additional practice p. 177 prob. 30-32

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Homework

• Create a study guide.
• There should be three questions for each aim. Be
  sure to include the solution step-by-step.
• In addition, include all key terms with their
  definitions.
• Go over previous homework problems and study the
  vocabulary.