1. Polynomial Functions Topic 2: Equations of Polynomial Functions

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

3. Explore… Complete the Explore Activity (Parts A through D) in your workbook before moving on.

4. You Should Notice… The constant term in a polynomial function is equal to the polynomial function’s y-intercept. The leading coefficient in any polynomial function is related it’s 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.

5. You Should Notice… The 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.

6. Information The 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.

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

8. Example 1 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. 11 y = -5 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  Q1 Q2  Q4 Linear equations go from Q3  Q1 (positive lead coefficient) or Q2  Q4 (negative lead coefficient).

9. Example 1 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).

10. Example 1 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).

11. Example 2 Match each graph with the correct polynomial function. Justify your reasoning. Matching polynomial functions to their graphs

12. Example 2 Matching 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.

13. Example 2 Matching 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.

14. Example 3 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. It’s minimum must be at y = -2, and it must have a y-intercept at 4.

15. Example 3 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.

16. Need to Know The 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.

17. 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.

18. Need to Know Linear and cubic polynomial functions have similar end behaviour.

19. Need to Know Quadratric polynomial functions have unique end behaviour. You’re ready! Try the homework from this section.