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 Identifying a polynomial function Identify which of the following functions is a polynomial function. Explain. a) b) c) 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…) Identifying a polynomial function Identify which of the following functions is a polynomial function. Explain. d) e) f) YES. YES. NO, since the variable has a negative exponent.

Example 2 Identifying the degree of a polynomial function 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) Degree 1 Linear Function Degree 2 Quadratic Function Degree 0 Constant Function

Example 2 (continued…) Identifying the degree of a polynomial function 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) 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 graph’s 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

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 For more practice with domain and range go to http://goo.gl/EwBax Information domain: the set of all possible x­-values range: the set of all possible y-values For example: The domain is the set of all real numbers since this graph extends all the way to the left and all the way to the right. {x|x∈R} The range is the set of all real numbers from 4 and down, since this graph’s highest point is 4, and it continues down forever. {y|y≤4, y∈R}

Explore… Complete this activity in your book before continuing! Identifying characteristics of a constant function 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. Complete this activity in your book before continuing!

You should notice… Constant functions produce graphs that are horizontal lines. The domain is always {x|x∈R} The range is always equal to the constant function. For a function y = -3, the range is {y|y=-3}. The number of x-intercepts is always 0, except if the constant function is y = 0. In this case, there are an infinite number of x-intercepts since the graph is a horizontal line on the x-axis. There is always 1 y-intercept, equal to the constant function. The end behaviour is Q2  Q1 or Q3 Q4

Explore… Complete this activity in your book before continuing! Identifying characteristics of a linear function 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. Complete this activity in your book before continuing!

You should notice… Linear functions (degree 0) produce graphs of diagonal lines. The domain is always {x|x∈R} The range is always {y|y∈R}. There is always 1 x-intercept and 1 y-intercept. The y-intercept is equal to the constant. The graph always proceeds from Q3 Q1 (if the equation has a positive lead coefficient) or Q2Q4 (if the equation has a negative lead coefficient).

Explore… Complete this activity in your book before continuing! Identifying characteristics of a quadratic function 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. Complete this activity in your book before continuing!

You should notice… Quadratic functions (degree 2) produce graphs that are parabolas. (For this course) the domain is always {x|x∈R} The range is always determined by the graph’s maximum or minimum value. For a function with a maximum at y = 4, the range is {y|y≤4,y∈R}. The graph always proceeds from Q2 Q1 (positive lead coefficient) or Q3Q4 ((negative lead coefficient) There is always 1 turning point. There is always 1 y-intercept, equal to the constant. The number of x-intercepts can be 2, 1, or 0, depending on how the graph is positioned.

Explore… Complete this activity in your book before continuing! Identifying characteristics of a cubic function 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. Complete this activity in your book before continuing!

You should notice… Cubic functions produce graphs that are curves. The domain is always {x|x∈R} The range is always {y|y∈R}. The graph always proceeds from Q3 Q1 (positive lead coefficient) or Q2Q4 (negative lead coefficient). There can be 0 or 2 turning points. There is always 1 y-intercept, equal to the y-intercept. The number of x-intercepts can be 1, 2, or 3, depending on how the graph is positioned..

Example 3 Summarizing and analyzing the characteristics of graphs of polynomials How is the maximum possible number of x-intercepts related to the degree of any polynomial function? Do all polynomials of degree 0, 1, 2, or 3 have only one y- intercept? What is the y-intercept for any polynomial function? 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 Summarizing and analyzing the characteristics of graphs of polynomials 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. The domain is always {x|x∈R}. For linear and cubic functions the range is {y|y∈R}. For quadratic functions, the range is always related to the maximum or minimum. Since cubic functions continue to go in both directions (up and down), their turning points are not maximum or minimum values.

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

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

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

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

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

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

Example 6 Try it first! Reasoning about the characteristics of the graphs of polynomial functions 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 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

Need to Know

You’re ready! Try the homework from this section. Need to Know You’re 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