Download - Polynomial Functions Topic 1: Graphs of Polynomial Functions

Polynomial Functions Topic 1: Graphs of Polynomial Functions

I can describe the characteristics of a polynomial function by analyzing its graph. I can describe the characteristics of a polynomial function by analyzing its equation. I can match equations to their corresponding graphs.

Information A polynomial function is a function that consists of one or more terms added together. Each term consists of a coefficient, a variable, and a whole number exponent. Variables cannot have negative or fractional exponents The variable cannot be in the exponent, the demoninator, or under a radical sign

Information The equations of polynomial functions can be written in standard form.

Information In a polynomial function, The leading coefficient is the coefficient of the term with the highest exponent of x in a polynomial function in standard form. The constant term is the term in which the variable has an exponent of 0. The degree of a function is the greatest exponent of the function. Polynomial functions are named according to their degree: constant functions have degree 0, linear functions have degree 1 quadratic functions have degree 2 cubic functions have degree 3

Example 1 Identify which of the following functions is a polynomial function. Explain. a) b) c) Identifying a polynomial function NO, since there is a fractional exponent. NO, since there is a variable inside a radical. NO, since there is a variable in the denominator.

Example 1 (continued) Identify which of the following functions is a polynomial function. Explain. d) e) f) Identifying a polynomial function YES. NO, since the variable has a negative exponent.

Example 2 Write the terms in each of the following polynomial functions in descending order. Identify the degree and the name of each function. a) b) c) Identifying the degree of a polynomial function Degree 1 Linear Function Degree 2 Quadratic Function Degree 0 Constant Function

Example 2 (continued) Write the terms in each of the following polynomial functions in descending order. Identify the degree and the name of each function. d) e) f) Identifying the degree of a polynomial function Degree 3 Cubic Function Degree 2 Quadratic Function Degree 3 Cubic Function

Information The graphs of polynomial functions have many characteristics. The characteristics that will be explored in this topic are as follows The x-intercept is the x-value of the point where a function crosses the x-axis. The y-intercept is the y-value of the point where a function crosses the y-axis. The domain is the x-values for which the function is defined. The range is the y-values for which the function is defined. The end behaviour of a function is the description of the graphs behaviour at the far left and far right.

Information A turning point of a function is any point where the y- values of a graph of a function change from increasing to decreasing or change from decreasing to increasing. An absolute maximum is the greatest value in the range of a function. An absolute minimum is the least value in the range of a function. A local maximum is a maximum turning point that is not the absolute maximum. A local minimum is a minimum turning point that is not the absolute minimum.

Information

The upcoming slides contain exploratory activities that you should work through independently. To do so, you need to remember the following: x-intercepts are the places where the graph crosses the x- axis (the horizontal axis). y-intercepts are the places in which the graph crosses the y-axis (the vertical axis). These ones are the x-intercepts (x = -1 and x = 3) These ones are the y-intercepts (y = 3)

Information domain: the set of all possible x-values range: the set of all possible y-values For example: For more practice with domain and range go to http://goo.gl/EwBa x

Use technology to investigate the characteristics of the following constant functions. Set your windows to X:[-5, 5, 1] and Y: [-5, 5, 1]. Record your findings in the table in your workbook. Identifying characteristics of a constant function Complete this activity in your book before continuing! Explore

You should notice

Use technology to investigate the characteristics of the following linear functions. Set your windows to X:[-5, 5, 1] and Y: [-5, 5, 1]. Record your findings in the table in your workbook. Identifying characteristics of a linear function Complete this activity in your book before continuing! Explore

You should notice

Use technology to investigate the characteristics of the following quadratic functions. Set your windows to X:[-5, 5, 1] and Y: [-5, 5, 1]. Record your findings in the table below. Identifying characteristics of a quadratic function Complete this activity in your book before continuing! Explore

You should notice

Use technology to investigate the characteristics of the following cubic functions. Set your windows to X:[-5, 5, 1] and Y: [-5, 5, 1]. Record your findings in the table below. Identifying characteristics of a cubic function Complete this activity in your book before continuing! Explore

You should notice

Example 3 a) How is the maximum possible number of x-intercepts related to the degree of any polynomial function? b) Do all polynomials of degree 0, 1, 2, or 3 have only one y- intercept? What is the y-intercept for any polynomial function? Summarizing and analyzing the characteristics of graphs of polynomials The maximum possible number of x-intercepts is equal to the degree of the polynomial function. All polynomials of degree 0, 1, 2, or 3 have only one y-intercept, and it is equal to the constant term in the equation.

Example 3 c) What are the domain and range for all polynomial functions? d) Explain why some cubic polynomial functions have turning points but not maximum or minimum values. Summarizing and analyzing the characteristics of graphs of polynomials Since cubic functions continue to go in both directions (up and down), their turning points are not maximum or minimum values.

Example 3 e) How is the leading coefficient in any polynomial function related to the graph of the function? Summarizing and analyzing the characteristics of graphs of polynomials The leading coefficient determines a graphs end behaviour. Equations with positive lead coefficients increase to the right. Equations with negative lead coefficients decrease to the right.

Example 4 Determine characteristics of each function using its equation. Using an equation to determine characteristics of a graph Since these are degree 1 polynomials, they each have only 1 x-intercept. 01 y = -3 y = 1 0 turning points The y-intercept is equal to the constant term. 0 turning points Linear functions are straight lines so there are no turning points. Q3 Q4 Q2 Q4 Linear equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4 (negative lead coefficient).

Example 4 Determine characteristics of each function using its equation. Using an equation to determine characteristics of a graph Since these are degree 2 polynomials, they each have up to 2 x-intercepts. 22 y = 8 y = -6 1 turning point The y-intercept is equal to the constant term. 1 turning point Quadratic functions are parabolas so there are 1 turning point. Q3 Q4 Q2 Q1 Linear equations go from Q2 Q1 (positive lead coefficient) or Q3 Q4 (negative lead coefficient). EXTRA (not in workbook)

Example 4 Determine characteristics of each function using its equation. Using an equation to determine characteristics of a graph Since these are degree 3 polynomials, they can have up to 3 x-intercepts. 33 y = -10 y = 3 2 turning points The y-intercept is equal to the constant term. 2 turning points Cubic functions are curves that can have up to 2 turning points. Q3 Q1 Q2 Q4 Cubic equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4 (negative lead coefficient). EXTRA (not in workbook)

Example 5 Match each graph with the correct polynomial function. Justify your reasoning. Matching polynomial functions to their graphs

Example 5 Matching polynomial functions to their graphs This equation is a cubic function with a negative lead coefficient (goes from Q2 Q4 and has a y-intercept of -2. It matches with graph v. This equation is a quadratic function with a positive lead coefficient (goes from Q2 Q1 and has a y-intercept of -2. It matches with graph iv. This equation is a cubic function with a positive lead coefficient (goes from Q3 Q1 and has a y-intercept of -2. It matches with graph i.

Example 5 Matching polynomial functions to their graphs This equation is a linear function with a negative lead coefficient (goes from Q2 Q4 and has a y- intercept of -3. It matches with graph vi. This equation is a quadratic function with a negative lead coefficient (goes from Q3 Q4 and has a y-intercept of -2. It matches with graph iii. This equation is a linear function with a positive lead coefficient (goes from Q3 Q1 and has a y-intercept of -3. It matches with graph ii.

Example 6 For each set of characteristics below, sketch the graph of a possible polynomial function. a)range: y-intercept: 4 Reasoning about the characteristics of the graphs of polynomial functions Try it first! Your answer may be different, but it must be a parabola that is opening upward. Its minimum must be at y = -2, and it must have a y-intercept at 4.

Example 6 For each set of characteristics below, sketch the graph of a possible polynomial function. b)range: turning points: one in quadrant III and one in quadrant I Reasoning about the characteristics of the graphs of polynomial functions Try it first! Your answer may be different, but it must be a cubic graph that goes from Q2 Q4. It has to have turning points in Q3 and Q1.

Need to Know A polynomial is a function that consists of one or more terms added together. Each term consists of a coefficient, a variable, and a whole number exponent. The leading coefficient is the coefficient of the term with the highest exponent of x in a polynomial function in standard form. The degree of a function is the greatest exponent of the function. It helps determine the shape of the graph of the function. The constant term is the term in which the variable has an exponent of 0.

Need to Know We can identify some characteristics of a polynomial function when the equation is written in standard form. The maximum number of x-intercepts is equal to the degree of the function. The maximum number of turning points is equal to one less than the degree of the function. The end behaviour is determined by the degree and leading coefficient. The y-intercept is the constant term.

Need to Know A turning point of a function is any point where the y- values of a graph of a function changes from increasing to decreasing or change from decreasing to increasing. An absolute maximum (minimum) is the greatest (least) value in the range. A local maximum (minimum) is a maximum (minimum) that is not an absolute maximum (minimum).

Need to Know

Youre ready! Try the homework from this section. Linear and cubic polynomial functions have similar end behaviour. if the lead coefficient is positive, the graph extends from quadrant III to quadrant I if the lead coefficient is negative, the graph extends from quadrant II to quadrant IV Quadratic polynomial functions have unique end behaviour. If the lead coefficient is positive, the graph extends from quadrant II to quadrant I If the lead coefficient is negative, the graph extends from quadrant III to quadrant IV

Download - Polynomial Functions Topic 1: Graphs of Polynomial Functions

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