Function Families Lesson 1-5

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?

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.

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) 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

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 Quadratic: y = x2 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

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

Graph of Quadratic Function

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 your constant. 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