Unit 4- Preview

Unit 4-Polynomial Functions

​-Preview-

1. To identify whether a function is even,odd, or neither using a graph.

2. To determine if a function is even, odd, or neither using a table and equation.

3. To describe the end behavior of a polynomial function of degree 3 or higher by using its degree and leading coefficient.

4. To sketch a rough graph of a polynomial function using zeros, multiplicity, and knowledge of end behavior.

5. To find the real and complex zeros of a polynomial equation of degree 3 or higher in factored form.

6. To find the real and complex zeros by factoring polynomial equations of degree 3 or higher.

7. To solve one-variable polynomial equations with suitable factorization within a real-world context.

We Will Learn

1. To identify whether a function is even,odd, or neither using a graph.

​What Should We Already Know?

• Given a graph, what is the axis of symmetry and end behaviors.

​What we will learn?

​

• Using what we know about the line of symmetry and end behaviors of graphs, we will be able to identify whether a function is even, odd, or neither.

What we will learn:

Multiple Choice

What is the axis of symmetry of the graph presented?

• x=-3

• x=1

• x=2

• x=-4

Multiple Choice

What are the end behaviors?

• As $$x\rightarrow+\infty,\ y\rightarrow-\infty$$

  As $$x\rightarrow-\infty,\ y\rightarrow-\infty$$

• As $$x\rightarrow+\infty,\ y\rightarrow+\infty$$

  As $$x\rightarrow-\infty,\ y\rightarrow-\infty$$

• As $$x\rightarrow-\infty,\ y\rightarrow+\infty$$

  As $$x\rightarrow-\infty,\ y\rightarrow-\infty$$

• As $$x\rightarrow+\infty,\ y\rightarrow-\infty$$

  As $$x\rightarrow+\infty,\ y\rightarrow-\infty$$

Drag and Drop

The axis of symmetry is​

Placeholder

. The left end behavior is ​​

Placeholder

​​ , and the right end behavior is​

Placeholder

​ . The function is ​​

Placeholder

.

Drag these tiles and drop them in the correct blank above

x=0

origin

even

odd

Drag and Drop

The axis of symmetry is ​

Placeholder

. The left end behavior is​

Placeholder

​ ​​ , and the right end behavior is ​

Placeholder

. The function is ​ ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

origin

odd

x=0

even

2. To determine if a function is even, odd, or neither using a table and equation.

​What Should We Already Know?

• Given a table, recognize axis of symmetry.

• Given a function, solve by substituting.

​What will we learn?

​

• Using what we know about axis of symmetry on a table, we can determine if the function is even, odd, or neither.

• Using what we know about solving by substituting, we can solve to determine if a function is even, odd, or neither.

What we will learn:

Dropdown

The function is (even/odd/neither)​

Select answer

because the table (shows/ does not show) ​

Select answer

symmetry about ​(y-axis/origin/ either one)

Select answer

.

Dropdown

The function is (even/odd/neither) ​ ​

Select answer

because the table (shows/ does not show) ​

Select answer

symmetry about ( origin/ y-axis/ either one)​

Select answer

.

Dropdown

The function is (even/ odd/ neither) ​ ​

Select answer

because the table (shows/ does not show) ​​

Select answer

symmetry about (y-axis/ origin/ either one) ​​

Select answer

.

Multiple Choice

$$f\left(x\right)=x^2+1$$

$$g\left(x\right)=9$$

What is $$f\left(g\left(x\right)\right)$$ ?

• 81

• 82

• 19

• 10

Multiple Choice

Determine whether each function is even, odd, or neither using the equation.

$$f\left(x\right)=x^3-2x$$

• even

• odd

• neither

Multiple Choice

Determine whether each function is even, odd, or neither using the equation.

$$f\left(x\right)=7x^4-1$$

• even

• odd

• neither

3. To describe the end behavior of a polynomial function of degree 3 or higher by using its degree and leading coefficient.

​What Should We Already Know?

• Identify an exponent.

• Identify a coefficient.

• Identify x- intercepts and zeros.

​What will we learn?

​

• How identifying the degree and leading coefficient of a polynomial function will help describe the end behavior of function.

• How to describe the relationship between number of zeros and their multiplicity and degree of a polynomial function.
  

What we will learn:

Degree of polynomial: The degree of a polynomial in one variable is the greatest exponent of its variable.

Leading Coefficient: The coefficient of the term with the highest term.

Ex:

Multiplicity: Number of times a factor repeats in a polynomial.

Multiple Select

$$y=x\left(x+2\right)\left(x-2\right)$$

What are the zeros?

• x=0

• x=2

• x=-2

• x=1

Dropdown

$$f\left(x\right)=4x^2-16x+16$$



The leading coefficient of the function is ​

Select answer

​ ​ and the degree is ​

Select answer

​ .

Multiple Select

$$f\left(x\right)=4x^2-16x+16$$

What are the zero (s)?

• x=0

• x=4

• x=2

• x=1

4. To sketch a rough graph of a polynomial function using zeros, multiplicity, and knowledge of end behavior.

​What Should We Already Know?

• Identify end behavior.

• Find zeros of function.

​What will we learn?

​

• Using what we know about degree, zeros, and leading coefficient, how to sketch a polynomial function on a graph.

What we will learn:

Ex:

Match

Match each graph with its equation.

• $$2\left(x+2\right)^2$$

• $$2\left(x-2\right)\left(x+2\right)$$

• $$2\left(x-2\right)^2$$

5. To find the real and complex zeros of a polynomial equation of degree 3 or higher in factored form.

​What Should We Already Know?

• Given factored form, solve for zeros.

​What will we learn?

​

• Find real and complex zeros of a polynomial equation.

What we will learn:

Complex zeros: solutions of a graph that are not visible on a graph. They are imaginary.

Multiple Choice

$$\left(4x^2-16\right)\left(x^2+9\right)$$

How many complex zeros are there?

• 2

• 3

• 1

• 4

6. To find the real and complex zeros by factoring polynomial equations of degree 3 or higher.

​What Should We Already Know?

• Factor by grouping.

• Identify perfect sum/ difference of cubes and squares and factor them.

​What will we learn?

​

• Through factoring find real and complex zeros of a polynomial equation.

What we will learn:

Factor to solve. (Grouping)

Ex:

Multiple Choice

Factored form of:

$$16x^4-25=0$$

• $$\left(4x^2-5\right)^2$$

• $$\left(4x+5\right)\left(4x-5\right)$$

• $$\left(4x^2+5\right)\left(4x^2-5\right)$$

• $$\left(4x^2+5\right)^2$$

Multiple Choice

Factored form of:

$$x^3+2x^2-4x-8=0$$

• $$\left(x^2-4\right)\left(x+2\right)$$

• $$\left(x+2\right)\left(x+2\right)\left(x-2\right)$$

• $$\left(x+2\right)\left(x-2\right)$$

• $$\left(x-4\right)\left(x+2\right)$$

Multiple Choice

Factor polynomial to solve.

$$x^3+x^2-9x-9=0$$

• x=3, -3, -1

• x=-3i, 3i, -1

• x=-3, 3, 3i, -3i, -1

• x=-1, 3, -3i

7. To solve one-variable polynomial equations with suitable factorization within a real-world context.

​What Should We Already Know?

• Identify and interpret parts of an equation in terms of real-world context.

​What will we learn?

​

• Solve one-variable polynomial equations in a real-world context.

What to look for in a real-world context...

• The function (Normally given)

• The zeros (Would solve for by factoring)

• Representation of zeros context (Based on the zeros)

What we will learn: