End Behavior and Symmetry

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1 Algebra 2 Interval Notation Name: Date: Block:

2 X Characteristics of Polynomial Functions Lesson Opener: Graph the function using transformations then identify key characteristics listed below. 1. y x A. Parent function: a, h, k 10 Y Table 1 Table Real zero(s): Domain: y-intercept: Range: Vertex: Interval of Increase: Axis of Symmetry: Interval of Decrease: Extrema:

3 End Behavior and Symmetry End behavior describes what the graph of the function does (y-values) as x goes to or. If a polynomial has Even degree, positive lead coefficient: both ends are up o as x, f ( x) Example: o as x, f ( x) Even degree, negative lead coefficient: both ends are down o as x, f ( x) Example: o as x, f ( x) Odd degree, positive lead coefficient: opposite end behavior with left end down and right end up o as x, f ( x) Example: o as x, f ( x) Odd degree, negative lead coefficient: opposite end behavior with left end up and right end down o as x, f ( x) Example: o as x, f ( x)

4 * Even/Odd/Neither and Symmetry Even functions have symmetry with respect to the y-axis Graphing Example: It s easy to determine if a polynomial is an even function: all the exponents are even If you have a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example: Odd functions have symmetry with respect to the origin Graphing Example: It s easy to determine if a polynomial is an odd function: all the exponents are odd Remember, if there is a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example: Neither means that the graph might have symmetry but neither with respect to the y-axis nor the origin. Graphing Example: It s easy to determine if a polynomial is a neither function: there is a mixture of even and odd exponents Remember, if there is a constant term, the exponent is zero and in this case it s considered an even term (NOT ODD) Function Example:

5 State the characteristics listed based on the graph of each function. 2 1.) f x x 2x g x x 5x 4 2.) 4 2 End behavior: as x, f ( x) End behavior: as x, f ( x) as x, f ( x) Is the function odd, even, or neither? Explain. as x, f ( x) Is the function odd, even, or neither? Explain.

6 h x x x 3.) 3 j x x 2x 3x 4.) 3 2 End behavior: as x, f ( x) End behavior: as x, f ( x) as x, f ( x) Is the function odd, even, or neither? Explain. as x, f ( x) Is the function odd, even, or neither? Explain.

7 Characteristics Notes Example 1: Odd Degree Identify the characteristics listed below for the given polynomial graph. A. Intercepts (all) x-intercepts: Points on the graph that are on the x-axis. They are also the real zeros of the function. A function may have no x-intercepts, one x-intercept, or several (up to the degree of the function). y-intercept: The single point on the graph that s on the y-axis. Every polynomial graph will have exactly one y-intercept. B. Relative and absolute extrema (specify) Extrema are all about the y-values. The absolute maximum is the highest y-value on a graph. The absolute minimum is the lowest y-value on the graph. An even degree function will have ONE absolute maximum or minimum, it cannot have both an absolute maximum AND minimum. A relative maximum is the highest y-value in that part of the graph, but not the highest of all the graph. A relative minimum is the lowest y-value in that part of the graph, but not the lowest of all the graph. A graph may have several relative maximums or minimums, or it may not have any at all. C. Domain: The domain of a function refers to all possible values of x (real numbers) that have a corresponding value of y. Like asking: What x-values am I allowed to choose? The domain of any polynomial function is all real numbers. This is written in interval notation as,. D. Range: The range of a function refers to any y-values found on the graph. The range of any odd degree function will always be all real numbers, or,. The range of an even degree function with a positive leading coefficient (opens up) will be abs min,. The range of an even degree function with a negative leading coefficient (opens down) will be,abs max.

8 E. Intervals of increase or decrease describe the behavior of the function (y-values) for each section of the domain which is divided by the maximum or minimum points (turning points). Intervals of increase or decrease are always stated using x-values. F. End Behavior As x, f x As x, f x G. Is the function odd, even, or neither? EXPLAIN. If a function has an odd degree AND is symmetric to the origin, then it s an odd function. If a function has an even degree AND is symmetric to the y-axis, then it s an even function. If a function is not symmetric to the origin or the y-axis it is neither (an odd or even function).

9 Example 2: Even Degree Identify the characteristics listed below for the given polynomial graph. A. Intercepts (all) x-intercepts: Points on the graph that are on the x-axis. They are also the real zeros of the function. A function may have no x-intercepts, one x-intercept, or several (up to the degree of the function). y-intercept: The single point on the graph that s on the y-axis. Every polynomial graph will have exactly one y-intercept. B. Relative and absolute extrema (specify): Extrema are all about the y-values. The absolute maximum is the highest y-value on a graph. The absolute minimum is the lowest y-value on the graph. An even degree function will have ONE absolute maximum or minimum, it cannot have both an absolute maximum AND minimum. A relative maximum is the highest y-value in that part of the graph, but not the highest of all the graph. A relative minimum is the lowest y-value in that part of the graph, but not the lowest of all the graph. A graph may have several relative maximums or minimums, or it may not have any at all. Domain: The domain of a function refers to all possible values of x (real numbers) that have a corresponding value of y. Like asking: What x-values am I allowed to choose? The domain of any polynomial function is all real numbers. This is written in interval notation as,. Range: The range of a function refers to any y-values found on the graph. The range of any odd degree function will always be all real numbers, or,. The range of an even degree function with a positive leading coefficient (opens up) will be abs min,. The range of an even degree function with a negative leading coefficient (opens down) will be,abs max.

10 E. Intervals of increase or decrease describe the behavior of the function (y-values) for each section of the domain which is divided by the maximum or minimum points (turning points). Intervals of increase or decrease are always stated using x-values. F. End Behavior As x, f x As x, f x G. Is the function odd, even, or neither? EXPLAIN. If a function has an odd degree AND is symmetric to the origin, then it s an odd function. If a function has an even degree AND is symmetric to the y-axis, then it s an even function. If a function is not symmetric to the origin or the y-axis it is neither (an odd or even function).

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