Determining Functions From Graphs To be a function, the graph
must pass the vertical line test. When a vertical line passes
through the graph, it should only touch one point at a time
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Example
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Describing Graphs of Linear Functions Positive Slope =
IncreasingNegative Slope = Decreasing
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Linear and Proportional Relationships All graphs of linear
proportional relationships are functions because they form a
straight line. Proportional: Straight line through (0, 0)Linear:
Straight line Function
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Nonlinear Relationships Many nonlinear relationships are
functions, but a graph or table may be needed to be sure. Function
Not a Function
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Notes- Functions from Tables A function is when each input
(x-value) corresponds to exactly one output (y- value) In other
words, when you substitute (x) into an equation there is only one
possible answer (y)
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Identifying Functions From a Table Every x input can have only
one corresponding output.
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Example xy 12 26 34 42 58 xy 12 26 14 42 58 FunctionNot a
function
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Example xy 12 26 34 42 58 xy 12 26 14 42 58 FunctionNot a
function Two different outputs for the same input
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Try- is this a function? xy 110 28 36 28 52 xy 18 24 30 44 58
xy 12 26 34 42 30
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Lets Graph One to See Why Each x Must Have a Unique y xy 00 24
34 42 31
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Lets Graph One to See Why xy 00 24 34 42 31
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For the following set of points, determine if the relationship
is a function 1)(-2, 3); (4, 2); (-3, 2); (4, 0) 2)(1, 4); (-3, 5);
(1, 4); (-2, 5); (3, 5) 3)(-5, 4); (4, -5); (-4, 5); (5, 4)
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Determine if the following is a function y = 2x y = 3x + 4
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Nope
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Determine if the following is a function by completing the
table and graphing y = x - 2 xy 0 1 2 -2
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Determine if the following is a function by completing the
table and graphing y = x - 3 xy 0 1 2 -2
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Writing the Rule for a Function
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Writing the Rule You need to look at the inputs and outputs in
the table to find a way to get from x to y that works for all
points. May be addition, subtraction, multiplication, or a
combination Write in the form y = mx + b
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Find the rule y = x + 3
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Find the Rule
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y = 3x
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Find the Rule What does the changing of signs tell us about the
rule?
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Find the Rule y = 2x + 2 How does is the value x = 0
helpful?
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Find the Rule A good trick is to find the difference or change
in x and y. That tells us what we are multiplying by
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Find the Rule 1 3 When the x value increases by 1, the y value
increases by 3. This tells us that x is being multiplied by 3 y =
3x ___
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Find the Rule
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Closure Get up and find a new partner Write the rule for the
following:
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Writing the Rule Given Two Points
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Rate of Change
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Find the rate of change The linear function goes through the
points (2, 4) and (4, 8)
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Find the rate of change The linear function goes through the
points (-3, 2) and (6, -1)
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Find the rate of change The linear function goes through the
points (-3, -5) and (-1, 3)
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Writing a Linear Function From Two Points This is a skill we
need to revisit. Find the rate of change (slope) Find the y
intercept (initial position) by substituting one coordinate pair
into y = mx + b
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Write the equation of a linear function that goes through the
points (-1, 1) and (1, 5)
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Write the equation of a linear function that goes through the
points (-4, 1) and (4, -3)
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Are You Serious Right Now?
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Carnival Amber and Mark went to the carnival on the same day.
There is a flat fee to enter, and all games are the same price.
Mark played 7 games and spent $12 (7, 12) and Amber played 11 games
and spent $16 (11, 16). What is the rate of change (how much is
each game)? How much was the entrance into the carnival?
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Kayaking Max and Ryder both rented kayaks and equipment on the
same day from the same company for a different length of time. The
company charges a flat fee to rent equipment and an hourly rate for
the kayaks. Max rented the kayak for 3 hours and paid $52. Ryder
rented the kayak for 7 hours and paid $112. What is the rate of
change (cost for one hour kayak rental)? How much was the equipment
rental?
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Cell Phone Bill Marcy recently signed up for a cell phone plan
and has no idea how much she is paying per minute, but knows that
her bill consists of a monthly fee and a cost per minute. She
looked at her bills from the last two months and found that she
used 500 minutes and paid $75 one month (500, 75) and she used 750
minutes and paid $100 the other month (750, 100). What is the rate
of change (cost per minute)? What is the monthly fee?
