Day _____ Notes: Parent Functions and Transformations ...
Day _____ Notes: Parent Functions and Transformations Pre-Calculus/Trigonometry Note: Start to memorize what each parent function looks like and its characteristics. A family of functions is a group of functions with graphs that display one or more similar characteristics. A parent function is the simplest of the functions in a family. This is the function that is transformed to create other members in the family of functions. In this lesson, you will study some of the most commonly used parent functions. You should already be familiar with some of them, and it is very important that you are familiar with all of them for Calculus. Function #1: Constant Function Equation: 𝑓(𝑥) = 𝑐 where c is any real number. Graph: Horizontal Line through c.
Domain: (−∞, ∞) Range: [c] Intercepts: no x-intercepts; y-intercept (0, c)
Function #2: Identity Function Equation: 𝑓(𝑥) = 𝑥 it passes through all points with the coordinates (a, a) Graph: Line that goes through the origin (direct variation)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Function #3: Quadratic Function Equation: 𝑓(𝑥) = 𝑥 2 Graph: U-shaped, vertex is at the origin, called a parabola
Domain: (−∞, ∞) Range: [0, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Y-axis (Even Function) Decreasing: (−∞, 0) Increasing: (0, ∞) Function #4: The Cubic Function Equation: 𝑓(𝑥) = 𝑥 3 Graph: Inflection point at (0, 0)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Function #5: The Square Root Function Equation: 𝑓(𝑥) = √𝑥 Graph: would not be a function if you included negative roots as well; inverse of the quadratic function
Domain: 𝑥 ≥ 0 Range: 𝑦 ≥ 0 Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: None (Neither Even nor Odd) Increasing: [0, ∞) Function #6: Reciprocal Function 1
Equation: 𝑓(𝑥) = 𝑥 Graph: called a hyperbola
Domain: 𝑥 ≠ 0 Range: 𝑦 ≠ 0 Intercepts: None Symmetry: Origin (Odd Function) Decreasing: (−∞, 0)U (0, ∞)
Function #7: Absolute Value Function Equation: 𝑓(𝑥) = |𝑥| Graph: V-shaped; actually a piecewise function made-up of 𝑦 = 𝑥 and 𝑦 = −𝑥
Domain: (−∞, ∞) Range: [0, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Y-axis (Even Function) Decreasing: (−∞, 0) Increasing: (0, ∞) Function #8: Cube Root Function 3
Equation: 𝑓(𝑥) = √𝑥 Graph: Inverse of the Cubic Function; Inflection Point at (0, 0)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Transformations of a parent function can affect the appearance of the parent graph. Rigid transformations change only the position of the graph, leaving the size and shape unchanged. Nonrigid transformations distort the shape of the graph. A translation is a rigid transformation that has the effect of shifting the graph of a function either horizontally or vertically. A vertical translation of a function f shifts the graph of f up or down, while a horizontal translation shifts the graph left or right.
Example #1: Use the graph of 𝑓(𝑥) = |𝑥| to graph each function. a) 𝑔(𝑥) = |𝑥| + 4
b) 𝑔(𝑥) = |𝑥 + 3|
c)
𝑔(𝑥) = |𝑥 − 2| − 1
Example #2: Use the graph of 𝑓(𝑥) = 𝑥 3 to graph each function. a)
𝑔(𝑥) = 𝑥 3 − 5
c) 𝑔(𝑥) = (𝑥 + 2)3 + 4
b) 𝑔(𝑥) = (𝑥 − 3)3
Another type of rigid transformation is a reflection, which produces a mirror image of the graph of a function with respect to a specific line.
When writing the equation for a transformed function, be careful to indicate the transformations correctly. The graph of 𝑔(𝑥) = −√𝑥 − 1 + 2 is different from the graph of 𝑔(𝑥) = −(√𝑥 − 1 + 2)
Example #3: Graph the following transformations. a)
𝑔(𝑥) = −𝑥 2 + 2
b) 𝑔(𝑥) = √−𝑥 − 3
A dilation is a nonrigid transformation that has the effect of compressing (shrinking) or expanding (enlarging) the graph of a function vertically or horizontally.
Study Tip: Sometimes pairs of dilations look similar like vertical stretches and horizontal compressions. It is not possible to tell which dilation a transformation is from the graph. You must compare the equation of the transformed function to the parent function. Example #4: Identify the parent function and describe how the graph of 𝑔(𝑥) is produced. a)
1
𝑔(𝑥) = 4 𝑥 3
b) 𝑔(𝑥) = −(0.2𝑥)2
Example #5: Identify the parent function and describe how the graph of 𝑔(𝑥) is produced. You can use your graphing calculator to confirm your answers. a)
6
𝑔(𝑥) = 𝑥 + 3
b) 𝑔(𝑥) = −4𝑥 2
Domain: (−∞, ∞) Range: [c] Intercepts: no x-intercepts; y-intercept (0, c)
Function #2: Identity Function Equation: 𝑓(𝑥) = 𝑥 it passes through all points with the coordinates (a, a) Graph: Line that goes through the origin (direct variation)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Function #3: Quadratic Function Equation: 𝑓(𝑥) = 𝑥 2 Graph: U-shaped, vertex is at the origin, called a parabola
Domain: (−∞, ∞) Range: [0, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Y-axis (Even Function) Decreasing: (−∞, 0) Increasing: (0, ∞) Function #4: The Cubic Function Equation: 𝑓(𝑥) = 𝑥 3 Graph: Inflection point at (0, 0)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Function #5: The Square Root Function Equation: 𝑓(𝑥) = √𝑥 Graph: would not be a function if you included negative roots as well; inverse of the quadratic function
Domain: 𝑥 ≥ 0 Range: 𝑦 ≥ 0 Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: None (Neither Even nor Odd) Increasing: [0, ∞) Function #6: Reciprocal Function 1
Equation: 𝑓(𝑥) = 𝑥 Graph: called a hyperbola
Domain: 𝑥 ≠ 0 Range: 𝑦 ≠ 0 Intercepts: None Symmetry: Origin (Odd Function) Decreasing: (−∞, 0)U (0, ∞)
Function #7: Absolute Value Function Equation: 𝑓(𝑥) = |𝑥| Graph: V-shaped; actually a piecewise function made-up of 𝑦 = 𝑥 and 𝑦 = −𝑥
Domain: (−∞, ∞) Range: [0, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Y-axis (Even Function) Decreasing: (−∞, 0) Increasing: (0, ∞) Function #8: Cube Root Function 3
Equation: 𝑓(𝑥) = √𝑥 Graph: Inverse of the Cubic Function; Inflection Point at (0, 0)
Domain: (−∞, ∞) Range: (−∞, ∞) Intercepts: x-intercept and y-intercept: (0, 0) Symmetry: Origin (Odd Function) Increasing: (−∞, ∞)
Transformations of a parent function can affect the appearance of the parent graph. Rigid transformations change only the position of the graph, leaving the size and shape unchanged. Nonrigid transformations distort the shape of the graph. A translation is a rigid transformation that has the effect of shifting the graph of a function either horizontally or vertically. A vertical translation of a function f shifts the graph of f up or down, while a horizontal translation shifts the graph left or right.
Example #1: Use the graph of 𝑓(𝑥) = |𝑥| to graph each function. a) 𝑔(𝑥) = |𝑥| + 4
b) 𝑔(𝑥) = |𝑥 + 3|
c)
𝑔(𝑥) = |𝑥 − 2| − 1
Example #2: Use the graph of 𝑓(𝑥) = 𝑥 3 to graph each function. a)
𝑔(𝑥) = 𝑥 3 − 5
c) 𝑔(𝑥) = (𝑥 + 2)3 + 4
b) 𝑔(𝑥) = (𝑥 − 3)3
Another type of rigid transformation is a reflection, which produces a mirror image of the graph of a function with respect to a specific line.
When writing the equation for a transformed function, be careful to indicate the transformations correctly. The graph of 𝑔(𝑥) = −√𝑥 − 1 + 2 is different from the graph of 𝑔(𝑥) = −(√𝑥 − 1 + 2)
Example #3: Graph the following transformations. a)
𝑔(𝑥) = −𝑥 2 + 2
b) 𝑔(𝑥) = √−𝑥 − 3
A dilation is a nonrigid transformation that has the effect of compressing (shrinking) or expanding (enlarging) the graph of a function vertically or horizontally.
Study Tip: Sometimes pairs of dilations look similar like vertical stretches and horizontal compressions. It is not possible to tell which dilation a transformation is from the graph. You must compare the equation of the transformed function to the parent function. Example #4: Identify the parent function and describe how the graph of 𝑔(𝑥) is produced. a)
1
𝑔(𝑥) = 4 𝑥 3
b) 𝑔(𝑥) = −(0.2𝑥)2
Example #5: Identify the parent function and describe how the graph of 𝑔(𝑥) is produced. You can use your graphing calculator to confirm your answers. a)
6
𝑔(𝑥) = 𝑥 + 3
b) 𝑔(𝑥) = −4𝑥 2
