Functions, Symmetry, and the Parent Functions

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Presentation transcript:

Functions, Symmetry, and the Parent Functions Honors Precalculus January 19-20, 2016 Mrs. Agnew

Essential Question Essential Vocabulary How do you find the domain of a function? How do you determine the symmetry of a function? Essential Vocabulary Function Parent Function Symmetry Domain Even/Odd Function

Terminology Function: rule that assigns every x to exactly one y. One-to-One Function? Domain: all x values for a given relation Range: all y values for a given relation Four methods of representing a function: visually, algebraically, verbally, numerically Examples: Function from graph? Domain/Range from graph

Function Notation Useful in demonstrating how to evaluate a function for a given value of x Terminology: independent variable, dependent variable Guided Practice Page 69 #40 – 46 (Even), 70

Symmetry with Respect to X-Axis Test for x-axis symmetry f(x) = - f(x) If you replace y with –y, do you return the same equation? Examples… y2 = 3x – 4 |y| = 5x2 – 4 y4 = -2x + 6

Symmetry with Respect to Y-Axis Test for y-axis symmetry f(x) = f(-x) If you replace x with –x, do you return the same equation? Examples… x2 = 5y – 8 y = -3|x| + 7 y = x4 – 7x2 + 2

Symmetry with Respect to Origin Test for origin symmetry f(x) = - f(-x) If you replace y with –y AND x with -x, do you return the same equation? Examples… x4 = -3y2 + 11 x2 + y2 = 9 y = - x Examples…

Graphing Using Symmetry Symmetry can help us graph by using a graph’s “Mirror Image” Graphing with symmetry is really reflecting a graph over the origin, x-axis, or y-axis Examples

Even and Odd Functions A function is EVEN if it is symmetric to the y-axis. A function is ODD if it is symmetric to the origin. What would a function be called if it were symmetric about the x-axis? Examples…

Homework: 1/19/17 Page 69 #43, 45, 69, 85, 87, 88, 95 Worksheet Provided

Domain & Range Recall that the domain of a function includes all of the values of x for which the function is defined. “Problems” in Domain Denominators Square Roots Range is all of the possible values for y. Examples…

Parent Functions The parent function is the “easiest” graph of a group of functions. We will investigate parent functions for the following functions: 1. Linear 5. Greatest Integer 2. Absolute Value 6. Reciprocal 3. Quadratic & 7. Semicircle Square Root 8. Exponential & 4. Cubic & Logarithmic Cubic Root 9. Trigonometric

Homework: 1/20/17 Page 69 #43, 45, 69, 85, 87, 88, 95 Worksheet Provided