Parent functions Module 2 Lesson 4

What is a parent function? We use the term ‘parent function’ to describe a family of graphs. The parent function gives a graph all of the unique properties and then we use transformations to move the graph around the plane. You have already seen this with lines. The parent function for a lines is y = x. To move the line around the Cartesian plane, we change the coefficient of x (the slope) and add or subtract a constant (the y intercept), to create the family of lines, or y = mx + b. To graph these functions, we will use a table of values to create points on the graph.

Parent Functions we will explore Name Parent Function Constant Function f(x) = a number Linear f(x) = x Absolute Value f(x) = |x| Quadratic f(x) = x2 Cubic f(x)= x3 Square Root Cubic Root Exponential f(x) = bx where b is a rational number

Symmetry Many parent functions have a form of symmetry. There are two types of symmetry: Symmetry over the y-axis Symmetry over the origin. If a function is symmetric over the y-axis, then it is an even function. If a function is symmetric over the origin, then it is an odd function.

Examples Even Functions Odd Functions

Parent Functions Menu (must be done from Slide Show View) 1) Click on this Button to View All of the Parent Functions. or 2) Click on a Specific Function Below for the Properties. 3) Use the button to return to this menu. Constant Function Cubic Root Linear (Identity) Exponential Absolute Value Square Root Quadratic Cubic

f(x) = a where a = any # Constant Function All constant functions are line with a slope equal to 0. They will have one y-intercept and either NO x-intercepts or they will be the x-axis. The domain will be {All Real numbers}, also written as or from negative infinity to positive infinity. The range will be {a}.

Domain: Range: f(x) = 2 y – intercept: x – intercept: None Example of a Constant Function Domain: Range: f(x) = 2 y – intercept: x – intercept: None

Graph Description: Horizontal Line Graphing a Constant Function f(x) = 2 x Y -2 2 -1 1 Graph Description: Horizontal Line

Linear Function f(x) = x All linear functions are lines. They will have one y-intercept and one x-intercept. The domain and range will be {All Real numbers}, or .

f(x) = x + 1 y – intercept: Domain: Range: (0, 1) x – intercept: Example of a Linear Function f(x) = x + 1 y – intercept: Domain: Range: (0, 1) x – intercept: (-1, 0)

Graph Description: Diagonal Line Graphing a Linear Function f(x) = x + 1 x y -2 -1 1 2 3 Graph Description: Diagonal Line

Absolute Value Function f(x) = │x│ All absolute value functions are V shaped. They will have one y-intercept and can have 0, 1, or 2 x-intercepts. The domain will be {All Real numbers}, also written as or from negative infinity to positive infinity. The range will begin at the vertex and then go to positive infinity or negative infinity.

Domain: y – intercept: x – intercepts: (0, 0) & (-2, 0) Range: Example of an Absolute Value Function f(x) = │x +1│-1 Domain: Range: y – intercept: x – intercepts: (0, 0) & (-2, 0)

Absolute Value Function Graphing an Absolute Value Function f(x) = │x+1 │-1 x y -2 -1 1 2 Graph Description: “V” - shaped

Quadratic Function f(x) = x 2 All quadratic functions are U shaped. They will have one y-intercept and can have 0, 1, or 2 x-intercepts. The domain will be {All Real numbers}, also written as or from negative infinity to positive infinity. The range will begin at the vertex and then go to positive infinity or negative infinity.

y – intercept: (0, -1) Domain: x – intercepts: (-1,0) & (1, 0) Range: Example of a Quadratic Function f(x) = x 2 -1 y – intercept: (0, -1) Domain: x – intercepts: (-1,0) & (1, 0) Range:

Graph Description: “U” - shaped Graphing a Quadratic Function f(x) = x 2 -1 x y -2 3 -1 1 2 Graph Description: “U” - shaped

Cubic Function f(x) = x 3 All cubic functions are S shaped. They will have one y-intercept and can have 0, 1, 2, or 3 x-intercepts. The domain and range will be {All Real numbers}, or .

y – intercept: (0, 1) Domain: x – intercept: (1, 0) Range: Example of a Cubic Function f(x) = -x 3 +1 y – intercept: (0, 1) Domain: x – intercept: (1, 0) Range:

Graph Description: Squiggle, Swivel Graphing a Cubic Function f(x) = -x 3 +1 x y -2 9 -1 2 1 -7 Graph Description: Squiggle, Swivel

Square Root Function f(x) = x All square root functions are shaped like half of a U. They can have 1 or no y-intercepts and can have 1 or no x-intercepts. The domain and range will be from the vertex to an infinity. .

y – intercept: (0, 2) Domain: x – intercept: none Range: Example of a Square Root Function f(x) = x + 2 y – intercept: (0, 2) Domain: x – intercept: none Range:

Graphing a Square Root Function f(x) = x + 2 x y 2 1 3 4 9 5 16 6 Graph Description: Horizontal ½ of a Parabola

Cubic Root Function All cubic functions are S shaped. They will have 1, 2, or 3 y-intercepts and can have 1 x-intercept. The domain and range will be {All Real numbers}, or .

y – intercept: Domain: x – intercept: Range: Example of a Cubic Root Function y – intercept: Domain: x – intercept: Range:

x y -2 1.26 1 6 2 Graph Description: S shaped Graphing a Cubic Root Function x y -2 1.26 1 6 2 Graph Description: S shaped

Exponential Function Where b = rational number All exponential functions are boomerang shaped. They will have 1 y-intercept and can have no or 1 x-intercept. They will also have an asymptote. The domain will always be , but the range will vary depending on where the curve is on the graph, but will always go an infinity.

y – intercept: Domain: x – intercept: none Range: Example of an Exponential Function f(x) = 2x y – intercept: Domain: x – intercept: none Range:

Graph Description: Backwards “L” Curves Graphing an Exponential Function f(x) = 2x x y -2 0.25 -1 0.5 1 2 4 Graph Description: Backwards “L” Curves