### 1-5 Function Families

```Lesson 1-5
Warm-up
 F(x) = 3x + 3
G(x) = x/3 - 1
F(6)
2. G(21)
3. F(-4)
4. G(-9)
5. F(0)
6. G(3)
Did you notice any relationship between the F functions
and the G functions?
1.
Warm-up
 Without looking back at your notes, define domain and range in
 Using your definitions, what is the domain and range of the
following graph? Assume that it doesn’t continue past this picture.
Warm-up
 For the following graph, find domain, range,
maximum, minimum, zeros (roots), y-intercepts,
intervals of increase and decrease, and the end
behavior.
What is a function family?
A function family is a group of functions that all share
the same characteristics. For example, all lines share
the characteristics that they have a domain and range
of all real numbers, they are continuous, and they have
a constant rate of change.
Important Definitions
 X-intercepts/roots – any location where the value (output)

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

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of the equation is equal to 0. In a graph, this is where the
graph crosses the x-axis
Y-intercepts – when the value of x = 0, we find our yintercept. In a graph, this is where the graph crosses the yaxis.
Domain – all possible x-values
Range – all possible y-values
Maximum – the ordered pair of the highest point on the
graph
Minimum – the ordered pair of the lowest point on the
graph
Important Definitions
 Increasing intervals – the x-values of the graph
between which the graph is going UP.
 Decreasing intervals – the x-values of the graph
between which the graph is going DOWN.
 Constant interval – the x-values of the graph between
which the graph is a STRAIGHT LINE.
 End Behavior – what is happening when the x-values
are becoming more negative or more positive out of
the graph.
Practice
 What is the domain, range, maximum, minimum, and
end behavior of each of the following?
1.
2.
3. (-3, 5), (-5, 2), (4, -3), (7, 0)
6 Function Families
 Linear: y = x
 Cubic: y = x3
 Absolute Value: y = |x|
 Square root: y = √x
 Rational: y = 1/x
Linear Functions
 Characteristics of a linear function
 Of the form y = x
 Domain: all real numbers
 Range: all real numbers
 Will have one root (x-intercept) and one y-intercept
 Has no maximum or minimum value
 Entire function is increasing
 End behavior in opposite directions
Graph of Linear Function
 Characteristics of a quadratic function (parabola)
 Of the form y = x2
 Domain: all real numbers
 Range: y ≥ 0 for parent graph.
 Minimum of 0 at the vertex in the parent graph.
 Can have 0, 1, or 2 roots (x-intercepts) and 1 y-intercept.
Has 1 root in the parent graph – the vertex.
 End behavior in the same direction, up.
 Interval of decrease x < 0; Interval of increase x > 0
Cubic Functions
 Characteristics of a cubic function
 Of the form y = x3
 Domain: all real numbers
 Range: all real numbers
 Will have neither a minimum nor a maximum value.
 Has 1 x-intercept (root) and 1 y-intercept: the origin
(0,0)
 End behavior in opposite directions: to negative infinity
as x approaches negative infinity; to positive infinity as x
approaches positive infinity
 Interval of increase: all real numbers or (-∞, ∞)
Graph of Cubic Function
Absolute Value Functions
 Characteristics of an absolute value function
 Of the form y = |x|
 Domain: all real numbers
 Range: y ≥ 0 for parent graph.
 Will have a minimum at the vertex: (0, 0)
 Has 1 root (x-intercept) and 1 y-intercept: (0, 0)
 End behavior in the same direction, up.
 Interval of decrease: x < 0; Interval of increase: x > 0
Graph of Absolute Value Function
Square root Functions
 Characteristics of an absolute value function
 Of the form y = √x
 Domain: x ≥ 0 for the parent graph.
 Range: y ≥ 0 for parent graph.
 Minimum value at the vertex: (0, 0)
 1 root (x-intercepts) and 1 y-intercept: (0, 0)
 End behavior to positive infinity.
 Interval of increase: x > 0 or [0, ∞)
Graph of Square Root Function
Rational Functions
 Characteristics of a rational function
 Of the form y = 1/x
 Domain: x ≠ 0 for the parent graph.
 Range: y ≠ 0 for parent graph.
 Will have neither a maximum nor a minimum
 Has neither a root (x-intercept) nor a y-intercept in the
original function. Instead, has a vertical asymptote that
on the y-axis and a horizontal asymptote on the x-axis.
 End behavior to 0 on both sides of the graph.
 Interval of decrease: all real numbers except x ≠ 0 or
(-∞, 0) U (0,∞)
Graph of Rational Function
Transformations
 What happens when you add or subtract a
constant from a parent function?
 The function shifts up or down the amount of
 What happens when you make a parent
function negative?
 The function is reflected across the x-axis.
Example of Vertical Translation
 y = x2
y = x2 - 4
Example of Reflection
Does a vertical translation affect
our following characteristics?
 Domain
 Range
 X-Intercept
 Y-Intercept
 Maximum
 Minimum
 Interval of Increase
 Interval of Decrease
 End Behavior
Does a reflection affect our
following characteristics?
 Domain
 Range
 X-Intercept
 Y-Intercept
 Maximum
 Minimum
 Interval of Increase
 Interval of Decrease
 End Behavior
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