Exploring Factors and Graphs

Exploring
Factors and

Graphs

Unit 3 Activity 1

Open this desmos link

https://www.desmos.com/calculator/cej6o9opbn

Notice

-You see three straight line (linear) functions.

-In the equations list, you see 7 entries but only 3 have colored circles next to them.

The colored circle means that entry is being graphed.

-Function f(x) has a red circle next to it. There is a red line in the graph.

-Function g(x) has a blue circle next to it. There is a blue line in the graph.

-Function h(x) has a green circle next to it. There is a green line in the graph.

Click the red circle next to function f(x)

What happened in your graph?

If you are not sure, click the circle again and observe how the graph changes.

Multiple Choice

What happened in your graph as you clicked the red circle?

• One of the lines disappeared.

• Four lines appeared.

• I did not click the red circle.

Multiple Choice

f(x) is a linear function with one factor, (x-1). Look at your desmos tab. Pay attention to function f(x) - in red. What is the x-intercept for this function?

• x= -1

• x= 0

• x= 1

• x= -3

Multiple Choice

g(x) is a linear function with one factor, (x+3). Look at your desmos tab. Pay attention to function g(x) - in blue. What is the x-intercept for this function?

• x= -1

• x= 0

• x= 1

• x= -3

Multiple Choice

h(x) is a linear function with one factor, (x). Look at your desmos tab. Pay attention to function h(x) - in green. What is the x-intercept for this function?

• x= 0

• x= 1

• No x-intercept

Notice

Did you notice the factor in the function and the x-intercepts are closely related?

A factor of (x-1) has an x-intecept of x=1

A factor of (x+3) has an x-intercept of x=-3

X intercepts are also called zeros.

The reason for this is because x-intercepts can be found by setting the equation equal to zero.

For example, x -1 = 0

+1 +1

x = 1

Function Families We Have Studied

Linear

Looks like y=mx+b

Have one linear factor.

Have one x-intercept (zero)

Straight line

Quadratic

Always has an x-squared term

Has 2 linear factors

May have up to 2 x intercepts

U-shaped parabola

Look at your desmos tab.

Consider function f(x) and function g(x).

If you multiply them together, you create a new function m(x).

Turn off h(x). Turn on m(x).

Multiple Choice

What family of function does m(x) belong to?

• linear

• quadratic

• something else

Multiple Choice

What are the factors of m(x)? What are the zeros of m(x)?

• Factors: (x-1) and (x+3)

  Zeros: x= -1 and x= 3

• Factors: (x-1) and (x+3)

  Zeros: x= 1 and x= -3

Look at your desmos tab.

Consider function f(x) and function h(x).

If you multiply them together, you create a new function n(x).

Turn off g(x) and m(x). Turn on n(x).

Multiple Choice

What family of function does n(x) belong to?

• linear

• quadratic

• something else

Multiple Choice

What are the factors of n(x)? What are the zeros of n(x)?

• Factors: (x-1) and (x)

  Zeros: x= 1 and x= 0

• Factors: (x-1) and (x)

  Zeros: x= -1 and x= 1

What if we have more than two linear factors?

So far, we have only explored functions that have one or two linear factors but it is
possible to have more factors.

If you have three linear factors, we call this function cubic.

We can even go higher than that.

A general word we use to describe functions that are composed of linear factors is
POLYNOMIAL.

Linear functions, quadratic functions, cubic functions are all examples of
polynomials.

Multiple Choice

Which of the following is an example of a polynomial function?

• Linear

• Quadratic

• Cubic

• All of these are polynomials

Look at your desmos tab.

Recall function m(x) was created by multiplying function f(x) and function h(x) to
form a quadratic function.

What happens if you multiply function m(x) by another linear factor?

Turn off all functions. Turn on functions m(x) and p(x).

Multiple Choice

What family of function does p(x) belong to?

• linear

• quadratic

• cubic

Multiple Choice

What are the factors of p(x)? What are the zeros of p(x)?

• Factors: (x-1), (x+3) and (x+2)

  Zeros: x= 1 , x=-3 and x= -2

• Factors: m and (x+2)

  Zeros: x= -2

Look at your desmos tab.

Turn off all functions.

Turn on function n(x). Recall n(x) is a quadratic function and has two linear factors.

If we multiply n(x) by a new factor, (x-5) we will get function r(x).

Turn on function r(x).

Is function r(x) quadratic? Is function r(x) cubic?

Multiple Choice

Is function r(x) a polynomial?

• yes- it is quadratic

• yes- it is cubic

• no it is not a polynomial

Multiple Choice

Consider the polynomial graph provided here. State the zeros for the polynomial.

• This is cubic with 3 zeros.

  Zeros: x= 2 , x=-1 and x= -4

• This is cubic with 3 zeros.

  Zeros: x= -2, x=1 and x= 4

Multiple Choice

A polynomial has the following factors:

(x-4), (x+2) and (x)

What can you say about this polynomial?

• It is quadratic.

• It crosses the x-axis only 2 times.

• It has zeros at x = 4, x = -2 and x = 0

Fill in the Blanks

A quick rule to use to find zeros of a polynomial when you know the factor is to switch the sign of the factor. For example, a factor of (x+1) has a zero at x=-1. A factor of (x-4) has a zero at x=4. A factor of (x+3) has a zero at x =

Poll

Today you explored how factors of an expression are related to graphs. Rank your understanding.

• 3- I understand that every factor has a corresponding zero (x-intercept) and I know how to find the x-intercept without graphing.

• 2- I understand that every factor has a corresponding zero (x-intercept) and I know how to find the x-intercept but only with a graph.

• 1- I understand that every polynomial has factors and zeros but not really sure what these mean.