Chapter 2 Linear Relations and Functions

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Chapter 2Linear Relations and Functions

• BY
• FRANKLIN KILBURN
• HONORS ALGEBRA 2

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Summary Slide

• 2 1Relations Functions
• 2 1 Cont'd
• 2 1 Cont'd
• 2 2 LINEAR EQUATIONS
• 2 2 Cont'd

3
Summary Slide (cont.)

• 2 3 SLOPE
• 2 3 Cont'd
• 2 3 Cont'd
• 2 4WRITING LINEAR EQUATIONS
• 2 4 Cont'd

4
Summary Slide (cont.)

• 2 5Modeling Real-World Data Using Scatter
  Plots
• 2 5 Cont'd
• 2 6SPECIAL FUNCTION
• 2 6 Cont'd

5
Summary Slide (cont.)

• 2 7GRAPHING INEQUALITIES
• 2 7 Cont'd
• 2 7 Cont'd
• Examples of Boundaries

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2 1Relations Functions

• Ordered pairs can be graphed on a coordinate
  system. The Cartesian coordinate plane is
  composed of the x-axis (horizontal) and the
  y-axis (vertical), which met at the origin (0,0)
  and divide the plane into four quadrants.

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2 1 Cont'd

• A relation is a set of ordered pairs.
• The domain of a relation is the set of all first
  coordinates (x-coordinates) from all the ordered
  pairs, and the range is the set of all ordered
  coordinates from all second coordinates
  (y-coordinates).

• The graph of a relation is the set of points in
  the coordinate plane corresponding to the ordered
  pairs in the relation.
• A function is a special type of relation in which
  each element of the domain is paired with exactly
  one element of the range.

• A mapping shows how each member of the domain is
  paired with each member of the range

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2 1 Cont'd

• When an equation represents a function there are
  two sets of variables
• The independent variable is usually x, and the
  values make up the domain.
• A dependent variable usually y, has values which
  depend on x.

• A function where each element of the range is
  paired exactly one element of the domain is
  called a one-to-one function.
• Vertical line test if no vertical line
  intersects a graph in more than one point, then
  the graph represents a function

• The equations are often written in functional
  notation. Ex y2x1 can be written as f(x)2x1.
  The symbol f(x) replaces the y and is read f of
  x.

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2 2 LINEAR EQUATIONS

• A linear equation has no operations other than
  addition, subtraction, and multiplication of a
  variable by a constant.
• The variables may not be multiplied together or
  appear in a denominator.
• Does not contain variables with exponents other
  than 1.
• The graph is always a line.

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2 2 Cont'd

• Any linear equation can be written in standard
  form
• AxByC
• where A, B, and C
• are real numbers.

• A linear function is a function whose ordered
  pairs satisfy a linear equation. Any linear
  function can be written
• in the form f(x) mxb, where m and b are
  real numbers.

• The y-intercept is the point of the graph in
  which the y-coordinate crosses the y-axis.

• The x-intercept is the point of the graph in
  which the x-coordinate crosses the x-axis.

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2 3 SLOPE

• The slope of a line is the ratio of the changes
  in y-coordinates to the change in x-coordinates.
  Slope measures how steep a line is.
• A family of graphs is the group of graphs that
  displays one or more similar characteristics.
• The parent graph is the simplest of the graphs in
  a family

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2 3 Cont'd

• The slope of a line tells the direction in which
  it rises of falls
• If the line rises to the right, the slope is
  positive.
• If the line is horizontal, the slope is zero.
• If the line falls to the right, the slope is
  negative.
• If the line is vertical, the line is undefined.

• The rate
• of change measures
• how much
• a quantity changes on average,
• relative to
• the change
• in another quantity.

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2 3 Cont'd

• In a plane, non-vertical lines with the same
  slope are parallel. All vertical lines are
  parallel.
• In a plane, two oblique lines are perpendicular
  if and only if the product of their slopes is -1.

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2 4WRITING LINEAR EQUATIONS

• Slope
• intercept
• form is the
• equation of
• a line in the
• form ymxb,
• where m is
• the slope
• and b is the
• y - intercept.

• An equation in the form
• y 4/3 x - 7
• is the point slope form.

• The slope-intercept and point-slope forms can be
  said to find equations of lines that are parallel
  or perpendicular to given lines.

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2 4 Cont'd

• The point - slope form of the equation of a line
  is y-y1m(x-x1) where (x1,y1) are the
  coordinates of a point on the line and m is the
  slope of the line.

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2 5Modeling Real-World Data Using Scatter
Plots

• Data with two variables such as speed and
  Calories is called bivariate data.
• A set of bivarate date graphed as ordered pairs
  in a coordinate plane is called a scatter plot.
• A scatter plot can show whether there is a
  relationship between the data.

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2 5 Cont'd

• A scatter plot is a set of data graphed as
  ordered pairs in a coordinate plane.
• An equation suggested by the points of a scatter
  plot used to predict other points is called a
  prediction equation.
• Line of fit line that closely approximates a
  set of data

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2 6SPECIAL FUNCTION

• A step function is a function whose graph is a
  series of line segments.
• A greatest integer function is a step function,
  written as f(x)x, where f(x) is the greatest
  integer less than or equal to x.
• A constant function is a linear function in the
  form of f(x)b.

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2 6 Cont'd

• Identity function the function of 1(x)x
• A piecewise function is written using two or more
  expressions
• A constant function is a linear function in the
  form of f(x)b.

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2 7GRAPHING INEQUALITIES

• A linear inequality resembles
• a linear equation, but with an inequality
  symbol rather than
• an equal symbol. Ex ylt2x1
• is a linear inequality and y2x1 is the
  related linear equation.

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2 7 Cont'd

• A boundary is a region bounded when the graph of
  a system of constraints is a polygonal region.

• Graphing absolute value inequalities is similar
  to graphing linear equations. The inequality
  symbol determines whether the boundary is solid
  or dashed, and you can test a point to determine
  which region to shade.

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2 7 Cont'd

• A linear inequality resembles a linear equation,
  but with an inequality symbol rather than an
  equal symbol.
• Ex ylt2x1 is a linear inequality and y2x1
  is the related linear equation.

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Examples of Boundaries

• Example 1
• Dashed Boundary

• Example 2
• Solid Boundary