Algebra 1
Topic 4:
Relations &
Functions
Table of Contents
1.
2.
3.
4.
5.
6.
Using Graphs to Relate Quantities
Patterns and Linear Functions
Nonlinear Functions
Graphing a Function Rule
Key Features of Graphs
Formalizing Relations & Functions
Using Graphs to
Relate
Quantities
Miami Traffic
Which graph could show a car sitting at a stoplight?
How do you know?
Connecting Graphs to Stories
Each day several leaves fall from a tree. One day a gust of wind
blows off many leaves. Eventually, there are no more leaves on
the tree. Choose the graph that best represents the situation.
Let’s Practice…
The air temperature increased steadily for several hours and then
remained constant. At the end of the day, the temperature
increased slightly before dropping sharply. Choose the graph that
best represents this situation.
Let’s Practice…
Marcus and Janine made the
table shown below to represent
the difference between their ages
during different years.
Which graph matches the
information in the table?
Let’s Practice…
The table shows the total number of customers at a car wash after
1, 2, 3, and 4 days of its grand opening. Which graph could
represent the data shown in the table?
Let’s Practice…
This graph shows someone taking a walk in the neighborhood.
Describe what it shows by labeling each part.
Draw Your Own Graph…
A pelican flies above the water searching for
fish. Sketch a graph of its altitude from takeoff
from shore to diving to the water to catch a fish.
Patterns and
Linear Functions
Vocab
Independent Variable – A variable that doesn’t
depend on any other value.
– Generally, x is used for the independent
variable.
Dependent Variable – A variable whose value
depends on some other value.
– Generally, y is used for the dependent
variable.
Identify the Independent and
Dependent Variables…
1. A painter must measure a room before
deciding how much paint to buy.
2. The height of a candle decrease d centimeters
for every hour it burns.
3. A veterinarian must weight an animal before
determining the amount of medication.
Vocab
Input – Values of the independent variable.
Output – Values of the dependent variable.
Independent Variable Dependent Variable
Vocab
Function is a set of ordered pairs (x, y) so
that each x-value corresponds to exactly
one y-value.
Function Rule
Output
variable
Input
variable
Representing Linear Relationships
Step 1: Make a table.
Step 2: Look for a pattern in the table.
Step 3: Create an equation to represent the
pattern.
Step 4: Use the table to make a graph.
What’s the Function?
A DVD buyers club charges a $20 membership fee and $15 per
DVD purchased. The table below represents this situation.
Number of DVDs
purchased
x
0
1
2
3
4
5
Total cost ($)
y
20
35
50
65
80
95
Let’s Practice…
The table shows the amount of
water y in a tank after x minutes
of being drained.
1. Is the relationship function?
2. Describe the relationship
using words.
3. Write an equation for the
relationship.
Nonlinear
Functions
Nonlinear Function
Nonlinear Function: a function whose graph is
not a line or part of a line.
Let’s Practice…
Is the graph is linear or nonlinear?
A.
B.
Let’s Practice…
Is the graph is linear or nonlinear?
C.
D.
What’s Special About Nonlinear
Functions?!?!
• Linear functions have a constant relationship
between x and y.
• HOWEVER, nonlinear functions, the differences
between x-values and the corresponding yvalues are not the same.
• Yet, there still is a pattern….
Let’s Practice…
What’s the pattern??
Tell whether the function in the table has a linear
or nonlinear relationship.
Let’s Practice…
What’s the pattern??
Tell whether the function in the table has a linear
or nonlinear relationship.
Input
Output
1
2
2
5
3
11
Writing a Rule to Describe a
Nonlinear Function
Step 1: Make a table to organize the x and y values.
Step 2: Look for a pattern in the y-values and
identify a rule that produces the given y-value when
you substitute the x-value.
Step 3: Write and verify the function rule.
Let’s Practice…
Write a function rule for the nonlinear function:
x
y
1
2
2
4
3
8
4
16
5
32
Let’s Practice…
The ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and
(5, 25) represent a function. What is a rule
that represents this function?
Graphing a
Function Rule
Vocab
Discrete: A discrete function is one which may take on only
a countable number of distinct values such as 0, 1, 2, 3, 4,...
Continuous: A continuous random variable is one which
takes an infinite number of possible values.
• Continuous random variables are usually measurements.
Car Approaching Traffic Light
Discrete or Continuous?
Henry begins to drain a
water tank by opening a
valve.
Water Level
Water tank
Time
Jamie is taking an 8-week
keyboarding class. At the end of
each week, she takes a test to find
the number of words she can type
per minute.
Graphing a Line
1. Pick three x-values.
• One should be negative, one should be
Procedure: Graphing Functions
zero and one should be positive.
2. Plug in the x-values to the function to
determine y value.
3. Graph each ordered pair.
4. Connect the dots with a line.
Let’s Practice…
Graph y = 2x + 1.
X
Y
Let’s Practice…
Graph y = 4x.
X
Y
Let’s Practice…
Graph 15x + 3y = 9.
X
Y
Dog Years…
The relationship between human years and dog years is given by
the function y = 7x, where x is the number of human years.
Graph this function.
X
Y
Let’s Practice…
At a salon, Sue can rent a
station for $10.00 per day plus
$3.00 per manicure. The
amount she would pay each
day is given by f(x) = 3x + 10,
where x is the number of
manicures. Graph this function.
X
Y
Graphing Nonlinear
Functions
1. Pick 5 x-values.
• Negatives, positives and zero.
2. Plug in the x-values to the function to
determine y-value.
3. Graph each ordered pair.
4. Connect the dots with a line.
Let’s Practice…
Graph the function f(x) = |x| + 2.
X
Y
Let’s Practice…
Graph the function y = |x – 1|.
X
Y
Let’s Practice…
Graph the function y = x2.
X
Y
Let’s Practice…
Graph the function y = x2 + 1.
X
Y
Key Features of
Graphs
Key Features of Graph
Intercepts
Relative Extremes
Symmetry
Slope
Form
X- Intercepts
Y- Intercepts
Relative Maximum
Relative Minimum
y-axis
x-axis
y=x
Positive
Negative
Increasing
Decreasing
Intercepts
X-intercept: Where the graph crosses the x
axis. The y-value is zero.
Y-intercept: Where the graph cross the y axis.
The x-value is zero.
Let’s Practice…
What are the x-intercept and y-intercept?
Relative Extremes
Relative Maximum: highest point on the graph
around it (looks like a hill)
Relative Minimum: lowest point on the graph
around it (looks like a valley)
Let’s Practice…
What is a relative maximum
of this graph?
What is a relative minimum
of this graph?
Symmetry
x-axis: When flipped over the x-axis, the graph
looks the same
Symmetry
y-axis: When flipped over the y-axis, the graph
looks the same
Symmetry
y=x: When flipped over the line y=x, the graph
looks the same
Let’s Practice…
What kind of symmetry does the picture have?
Slope
Positive: the graph increases from left to right
Negative: the graph decreases from left to right
Negative
Positive
Let’s Practice…
Of lines A, B, and C,
which lines are positive?
Which lines are
negative?
B
A
C
Form
Increasing: The values of y get bigger from left to right
Decreasing: The values of y get smaller from left to right
Let’s Practice…
Name the internals
where the function is
increasing and
decreasing.
Formalizing
Relations and
Functions
Relation
Relation – A relationships that can be
represented by a set of ordered pairs.
Table
x
y
2
3
4
7
6
8
Example: Express the relation {(2, 3), (4, 7),
(6, 8)} as a table, as a graph, and as a
mapping diagram.
y
x
2
3
4
7
6
8
Domain & Range
Domain: The set of first coordinates (or xvalues) of the ordered pairs of a relation.
Range: The set of second coordinates (or
y-values) of the ordered pairs of a
relation.
Example: Finding Domain and
Range from a Graph
y = x2
Only
nonnegative yvalues appear
on the graph.
All x-values appear
somewhere on the graph.
For y = x2 the domain is all real numbers and the
range is y ≥ 0.
Let’s Practice…
Give the domain and range of the relation.
Let’s Practice…
Give the domain and range of the relation.
x
1
4
8
y
6
1
5
4
2
1
1
–4
–1
0
Functions
Function: a special type of relation that pairs
each domain value with exactly one range value.
All functions are relations, but all relations are
not functions.
Is This Relation a Function?
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
1. {(3, –2), (5, –1), (4, 0), (3, 1)}
2. {(-1, 5), (0, 2), (1, 3), (5, 7)}
3. {(3, 9), (4, -6), (0, 9), (-1, -5)}
Is This Relation a Function?
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
1. {(3, –2), (5, –1), (4, 0), (3, 1)}
D: {3, 4, 5}
R: {-2, -1, 0, 1}
No. The 3 connects to both the -2 and 1.
2. {(-1, 5), (0, 2), (1, 3), (5, 7)}
D: {-1, 0, 1, 5} R: {2, 3, 5, 7}
Yes, since each domain value does not repeat.
3. {(3, 9), (4, -6), (0, 9), (-1, -5)}
D: {-1, 0, 3, 4} R: {-5, -6, 9}
Yes, since each domain value does not repeat.
Is This Relation a Function?
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
–4
–8
4
5
2
1
Let’s Practice…
Give the domain and range of the relation. Tell
whether the relation is a function. Explain.
a. {(8, 2), (–4, 1),
(–6, 2),(1, 9)}
b.
Let’s Practice…
Give the domain and range of each relation. Tell
whether the relation is a function and explain.
Short Cut: Vertical Line Test
Let’s Practice…
Graph each equation.
Then tell whether the
equation represents a
function.
y = 3x – 2
X
Y
Let’s Practice…
Graph each equation. Then tell whether the
equation represents a function.
y = |x – 1|
X
Y
Function Notation
f(x) is a fancy way of saying “y”
Evaluating Functions & Finding
the Range
• We can evaluate a function and find the range
by plugging in!
Example:
Find the range: f(x) = 3x + 2 using (1, 2, 3, 4)
Let’s Practice…
Find the range:
1. f(x) = 2x -7 (-2, -1, 0, 1 , 2)
2. h(x) = x2; (-5, 0, 3, 4, 7)