1. Function Families
Lesson 1-5

2. Warm-up F(x) = 3x + 3 G(x) = x/3 - 1 F(6) G(21) F(-4) G(-9) F(0) G(3)
Did you notice any relationship between the F functions and the G functions?

3. Warm-up
Without looking back at your notes, define domain and range in your own words.
Using your definitions, what is the domain and range of the following graph? Assume that it doesn’t continue past this picture.

4. Warm-up
For the following graph, find domain, range, maximum, minimum, zeros (roots), y-intercepts, intervals of increase and decrease, and the end behavior.

5. 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.

6. Important Definitions
X-intercepts/roots – any location where the value (output) 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 y-intercept. In a graph, this is where the graph crosses the y-axis.
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

7. 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.

8. Practice
What is the domain, range, maximum, minimum, and end behavior of each of the following?
3. (-3, 5), (-5, 2), (4, -3), (7, 0)

9. 6 Function Families Linear: y = x Quadratic: y = x2 Cubic: y = x3
Absolute Value: y = |x|
Square root: y = √x
Rational: y = 1/x

10. 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

11. Graph of Linear Function

12. Quadratic Functions 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

13. Graph of Quadratic Function

14. 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 (-∞, ∞)

15. Graph of Cubic Function

16. 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

17. Graph of Absolute Value Function

18. 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, ∞)

19. Graph of Square Root Function

20. 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,∞)

21. Graph of Rational Function

22. Transformations
What happens when you add or subtract a constant from a parent function?
The function shifts up or down the amount of your constant.
What happens when you make a parent function negative?
The function is reflected across the x-axis.

23. Example of Vertical Translation
y = x2 y = x2 - 4

24. Example of Reflection

25. Does a vertical translation affect our following characteristics?
Domain
Range
X-Intercept
Y-Intercept
Maximum
Minimum
Interval of Increase
Interval of Decrease
End Behavior

26. Does a reflection affect our following characteristics?
Domain
Range
X-Intercept
Y-Intercept
Maximum
Minimum
Interval of Increase
Interval of Decrease
End Behavior