Ch 6 – Functions and Graphs
6.1 – Relations
Ordered Pair:
Independent Variable:
Dependent Variable:
Relation:
Domain:
Range:
Example:
Ordered Pair:
Relation:
Domain:
Range:
Three ways to represent a relation:
Example: Express the relation {(-4, 5), (-3, 2), (0, 1), (1, -1), (3, -2)} as a table
and as a graph. Then determine the domain and range.
Example: Express the relation shown on the graph as a set of ordered pairs and in a table. Then find the domain and
range.
Example: The table shows the population of New York City since 1920.
Determine the domain and range of the relation.
Graph the relation
During which decade was there the greatest increase in population?
6.2: Equations as Relations
Solution of an Equation in Two Variables:
Solution set:
Example: Which of the ordered pairs (0, 0), (1, 4), (2, 1) or (-1, 2) are solutions of y = -x + 3?
Example: Which of the ordered pairs (1, 1), (0, 2), (-2, 8) or (-1, -5) are solutions of y = -3x + 2?
Example: Solve y = 2x + 1 if the domain is {-2, -1, 0, 1, 3}. Graph the solution
set.
Example: Solve y = -4x if the domain is {-2, -1, 0, 1, 2}. Graph the solution set.
Example: Solve 6x + 3y = 12 if the domain is {-2, -1, 0, 1, 2}. Graph the solution set.
Example: Find the domain of y = 10 – 4x if the range is {-6, -2, 2, 6, 10}.
Example: Find the domain of y = -3x if the range is {-6, -3, 0, 3, 6}.
Example: The equation 2w + 2l = P can be used to find the perimeter, P, of a rectangle. Suppose a rectangle has a
perimeter of 20 inches. Find the possible dimensions of the rectangle given the domain values {1, 2, 4, 5}.
6.3: Graphing Linear Relations
Linear Equation –
Standard form for linear equations -
Linear Equation -
Example: Determine whether each equation is a linear equation. Explain. If an equation is linear, identify A, B, and
C when written in standard form.
4xy = 4
y=x
y=7
8 + y = 2x
-5z = 4x + 2y
y2 = 4
Example: Graph y = 2x - 1
Example: Graph 3x – y = 2
Example: Graph 3x + 2y = 4
Example: Graph y = -2. Describe the graph.
Example: Graph x = -2. Describe the graph.
Domain and range of lines
Domain:
Range:
Since a line extends forever in directions,
Unless the line is vertical or horizontal
Vertical:
Horizontal:
Example: Graph y = -3x.
Example: Graph y = 3x
Special Note
Example: Old Faithful is a famous geyser. The equation y = 14x + 27 can be used to predict its eruptions, where y is
the time until the next eruption (in minutes) and x is the length of the eruption (in minutes). Suppose Old Faithful
has an eruption that lasts 1.4 minutes. About how long will it be until the next eruption?
6.4: Functions
Function:
Example: Determine whether each relation is a function. Explain your answer.
Vertical line test:
Example: Use the vertical line test to determine whether each relation is a function.
Functional notation:
Example: If f(x) = x – 4, find each value.
f(2)
f(1/2)
f(c)
Example: If g(x) = 3x – 7, find each value.
g(-2)
g(1.5)
g(2h)
Example: If f(x) = 2x + 3, find each value.
f(4)
f(-½)
f(6a)
Example: Anthropologists use the length of certain bones of a human skeleton to estimate the height of the living
person. One of these bones is the femur, which extends from the hip to the knee. To estimate the height in
centimeters of a female with a femur of length, x, the function h(x) = 61.41 + 2.32x can be used.
What was the height of a female who femur measures 46 centimeters?
What was the height of a female whose femur measured 20 centimeters?
6.5: Direct Variation
Direct Variation General form:
Example: Determine whether the equation is a direct variation.
y = ½x
y = 2x + 1
y=x+2
y = -3x
Example: If an object that weighs 110 pounds on Earth weighs 18.7 pounds on the moon, how much would an object
weight on the moon if its weight on Earth is 275 pounds?
Example: An object that weighs 110 pounds on Earth weighs 18.7 pounds on the moon. How much would an object
weight on the moon if the weight on Earth is 150 pounds?
Example: The length of a trip varies directly as the amount of gasoline used. How many gallons of gasoline would be
needed for a 550-mile trip if a 66-mile trip used 3 gallons of gasoline? Hint: use l = kg
Example: How many gallons of gasoline would be needed for a 363-mile trip if a 72.6-mile trip used 3 gallons of
gasoline?
Example: Suppose y varies directly as x and y = 27 when x = 18. Find x when y = 15.
Example: Suppose y varies directly as x and y = 27 when x = 6. Find x when y = 45.
Example: If there are 16 ounces in 1 pound, how many ounces are in 12 pounds?
Example: How many pints are in 2.3 gallons if there are 8 pints in 1 gallon?
6.6: Inverse Variation
Inverse variation:
Example: The number of bricklayers needed to build a brick wall varies inversely as the number of days needed. If 4
bricklayers can build a brick wall in 30 hours, how long would it take 5 bricklayers to do it?
Example: The number of masons needed to build a block basement varies inversely as the number of days needed.
It 8 masons can build a block basement in 3 days, how long will 9 masons take to do it?
Example: The number of carpenters needed to frame a house varies inversely as the number of days needed to
complete the project. Suppose 5 carpenters can frame a house in 16 days. How many days will 8 carpenters take to
frame the house? Assume that they all work at the same rate.
Example: Suppose y varies inversely as x and y = -2 when x = -12. Find y when x = 8.
Example: Suppose y varies inversely as x and y = 3 when x = 12. Find x when y = 4.
Example: A student running at 5 miles per hour runs one lap around the school campus in 8 minutes. If a second
student takes 10 minutes to run one lap around the school campus, how fast is she running?