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# Module 2 lesson 4 notes

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• 1. PARENT FUNCTIONS Module 2 Lesson 4
• 2. 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 graphthesefunctions, wewill usea tableof values to createpoints onthegraph.
• 3. Parent Functions we will explore xy = Name Parent Function Constant Function f(x) = a numberLinear f(x) = xAbsolute Value f(x) = |x|Quadratic f(x) = x2 Cubic f(x)= x3 Square Root Cubic Root Exponential x 3 )( xxf = xxf =)(
• 4. 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.
• 5. Examples Even Functions Odd Functions
• 6. Parent Functions Menu (must be done from Slide Show View) Constant Function Cubic Root Linear (Identity) Exponential Absolute Value Square Root Quadratic Cubic 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.
• 7. Constant Function f(x) = a where a = any # 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}. ( ),−∞ ∞
• 8. Example of a Constant Function Domain: ( ),−∞ ∞ Range: 2y = f(x) = 2 x – intercept: y – intercept: ( )0, 2 None
• 9. Graphing a Constant Function Graph Description: Horizontal Line x Y -2 2 -1 2 0 2 1 2 2 2 f(x) = 2
• 10. 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 .( ),−∞ ∞
• 11. Example of a Linear Function ( ),−∞ ∞ f(x) = x + 1 Domain: ( ),−∞ ∞ Range:x – intercept: y – intercept: (-1, 0) (0, 1)
• 12. x y -2 -1 -1 0 0 1 1 2 2 3 Graph Description: Diagonal Line f(x) = x + 1 Graphing a Linear Function
• 13. 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. ( ),−∞ ∞
• 14. Example of an Absolute Value Function f(x) = │x +1│-1 Domain: ( ),−∞ ∞ Range:x – intercepts: (0, 0) & (-2, 0) y – intercept: ( )0, 0 ),1[ ∞−
• 15. x y -2 -2 -1 -1 0 0 1 1 2 2 Graph Description: “V” - shaped f(x) = │x+1 │-1 Graphing an Absolute Value Function
• 16. f(x) = x 2 Quadratic Function 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. ( ),−∞ ∞
• 17. f(x) = x 2 -1 Example of a Quadratic Function Domain: ( ),−∞ ∞ Range: ),1[ ∞−x – intercepts: (-1,0) & (1, 0) y – intercept: (0, -1)
• 18. f(x) = x 2 -1 x y -2 3 -1 0 0 -1 1 0 2 3 Graph Description: “U” - shaped Graphing a Quadratic Function
• 19. 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 . ( ),−∞ ∞
• 20. Example of a Cubic Function f(x) = -x 3 +1 Domain: ( ),−∞ ∞ Range:x – intercept: (1, 0) y – intercept: (0, 1) ( ),−∞ ∞
• 21. x y -2 9 -1 2 0 1 1 0 2 -7 Graph Description: Squiggle, Swivel f(x) = -x 3 +1 Graphing a Cubic Function
• 22. f(x) = x Square Root Function 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. .
• 23. f(x) = x + 2 Example of a Square Root Function Domain: Range:x – intercept: none y – intercept: (0, 2) ),0[ ∞ ),2[ ∞
• 24. x y 0 2 1 3 4 4 9 5 16 6 Graph Description: Horizontal ½ of a Parabola f(x) = x + 2 Graphing a Square Root Function
• 25. Cubic Root Function 3 )( xxf = 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 .( ),−∞ ∞
• 26. Example of a Cubic Root Function 3 2)( += xxf Domain: Range:x – intercept: y – intercept: ),( ∞−∞ ),( ∞−∞ )2,0( 3 )0,2(−
• 27. x y -2 0 0 1.26 1 1 6 2 Graph Description: S shaped 3 2)( += xxf Graphing a Cubic Root Function
• 28. Exponential Function ( ) x f x b= 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. ),( ∞−∞
• 29. f(x) = 2x Example of an Exponential Function Domain: Range:x – intercept: none y – intercept: ),( ∞−∞ ),0( ∞ )1,0(
• 30. f(x) = 2x x y -2 0.25 -1 0.5 0 1 1 2 2 4 Graph Description: Backwards “L” Curves Graphing an Exponential Function