# polynomial functions topic 2: equations of polynomial functions

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Topic 1: The Fundamental Counting Principle

Polynomial FunctionsTopic 2: Equations of Polynomial FunctionsI can describe the characteristics of a polynomial function by analyzing its equation.I can match equations to their corresponding graphs.ExploreComplete the Explore Activity (Parts A through D) in your workbook before moving on.

You Should NoticeThe constant term in a polynomial function is equal to the polynomial functions y-intercept.

The leading coefficient in any polynomial function is related its end behaviour. In a linear or cubic function, a positive lead coefficient means it proceeds from Q3 Q1 and a negative lead coefficient proceeds from Q2 Q4.In a quadratic function, a positive lead coefficient means it proceeds from Q2 Q1 and a negative lead coefficient proceeds from Q3 Q4.

You Should NoticeThe degree of a polynomial function equal to the maximum number of x-intercepts.The degree of a polynomial function is equal to 1 more than the maximum number of turning points.

InformationThe equation of polynomials can be written in standard form.

The leading coefficient in a polynomial function is the coefficient of the term with the highest exponent of x in a polynomial function in standard form.

InformationThe constant term is the term in which the variable has an exponent of 0. In other words, the constant term is the number that looks like it has no variable.

We can identify some characteristics of a polynomial function when the equation is written in standard form.

Example 1Determine 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.11y = -5y = 10 turning pointsThe y-intercept is equal to the constant term.0 turning pointsLinear functions are straight lines so there are no turning points.Q3 Q1Q2 Q4

Linear equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4 (negative lead coefficient).Example 1Determine 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.22y = 8y = -61 turning pointThe y-intercept is equal to the constant term.1 turning pointQuadratic functions are parabolas so there are 1 turning point.Q3 Q4Q2 Q1Linear equations go from Q2 Q1 (positive lead coefficient) or Q3 Q4 (negative lead coefficient).Example 1Determine 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.33y = -10y = 32 turning pointsThe y-intercept is equal to the constant term.2 turning pointsCubic functions are curves that can have up to 2 turning points.Q3 Q1Q2 Q4Cubic equations go from Q3 Q1 (positive lead coefficient) or Q2 Q4 (negative lead coefficient).Example 2Match each graph with the correct polynomial function. Justify your reasoning.Matching polynomial functions to their graphs

Example 2Matching polynomial functions to their graphs

This equation is a cubic function with a negative lead coefficient (goes from Q2 Q4 and a y-intercept of -2. It matches with graph vi.

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 ii.This equation is a cubic function with a positive lead coefficient (goes from Q3 Q1 and a y-intercept of -2. It matches with graph iv.

Example 2Matching polynomial functions to their graphs

This equation is a linear function with a negative lead coefficient (goes from Q2 Q4 and a y-intercept of -3. It matches with graph iii.This equation is a quadratic function with a positive lead coefficient (goes from Q2 Q1 and has a y-intercept of 1. It matches with graph v.This equation is a linear function with a negative lead coefficient (goes from Q2 Q4 and a y-intercept of -3. It matches with graph i.

Example 3For 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 3For 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 KnowThe equations of polynomials can be written in standard form.

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. In other words, the constant term is the number that looks like it has no variable.

Need to KnowWe 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 KnowLinear and cubic polynomial functions have similar end behaviour.

Need to KnowQuadratric polynomial functions have unique end behaviour.

Youre ready! Try the homework from this section.

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