Function Families

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? 1. 2. 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 ≥ 0for 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