Lesson 3.6 Classwork

Lesson 3.6 Classwork

SOLVE POLYNOMIAL EQUATIONS BY GRAPHING

The purpose of this assignment:

You will learn how to solve polynomial equations by graphing

You will practice using a graphing calculator to find intersections, maximums, and minimums

This assignment will be a daily grade

You will need the worksheets from class to answer some of these questions. Remote learners, these pages are in google classroom.

THE QUESTION ON THE NEXT SLIDE IS ABOUT PAGE 23

Multiple Choice

Describe the end behavior of the polynomial that we graphed on page 23. (it says 'Polynomial Review' at the top)

• both ends point up

• both ends point down

• down left, up right

• up left, down right

Multiple Choice

How do you know the y-intercept of a function? (like the one on page 23)

• It will always be the leading coefficient

• It will be the highest exponent

• It will be the constant term at the end

Multiple Choice

When finding roots of a polynomial, when can you use the quadratic formula?

• when you have a cubic function

• when you have a quadratic function

• when you get tired

• never!

Multiple Choice

When you graph the 2 sides of a polynomial equation, where do you look for the solutions?

• y-intercepts

• x-intercepts

• where the graphs intersect

• under the couch

COMPLETE EXAMPLE 2 (IF YOU HAVEN'T)

ANSWERS AT THE BOTTOM OF THE PAGE:

*To find the REAL solutions of an equation, rewrite the equation as a SYSTEM

*The solution is the x value of the intersection. Round to the nearest 100th

The question on the next slide is about example number 2. Be sure you have it done before moving on.

Multiple Choice

Which of the following is a solution to Example 2 on page 24?

• 1.41

• 2.75

• -3.15

• 1.28

Below, is the correct equation for the cereal box volume problem (#2) on page 25. Check your answer.

x³ + 6x² - 40x = 192

Multiple Choice

What are the 3 solutions to the cereal box system problem?

• 6, -4, 2

• -8, -4, 6

• 1, 5, -10

• 3, 7, 9

Here is the answer for #6 on page 25 (the application problem page)

x³ + 6x² - 40x - 192

Multiple Choice

When you graph the function: x³ + 6x² - 40x - 192, what are the x-intercepts?

• -8, -4, 6

• 3, 7, 9

• -2, 6, 8

• 0, 5, 7

The notes on the following slide are for page 26. You could also just write these on notebook paper.

Absolute Maximum: highest point of the entire graph.

Relative Maximum: a point that is higher than the points around it (like a mountain peak)

Absolute Minimum: LOWEST point of the entire graph.

Relative Minimum: a point that is LOWER than the points around it (like a valley)

Sketch this example on your paper

Every graph does not have to have a max AND a min.

A max or min is a POINT (a number). It CANNOT be infinity!!!

This graph has NO ABSOLUTE MAXIMUM, because the arrows point UP forever.

This graph only has RELATIVE EXTREMA. NO ABSOLUTES! (because there are arrows pointing UP AND DOWN forever)

MAX & MIN values are the Y-values of the points. The relative maxes are 2 and 1 -------->

Multiple Select

What are the relative mimimums of the graph? (choose both)

• -3

• 2

• -1

• -5

Multiple Choice

What is the absolute maximum???

• THERE IS NOT AN ABSOLUTE MAXIMUM

• 1

• 2

• 3

Multiple Choice

(T/F) The function shown has an Absolute Minimum.

• True

• False

Multiple Choice

Which of the following does this graph not show?

• Absolute Max

• Absolute Min

• Relative Max

• Relative Min

Multiple Choice

What are the Relative Maxima of this graph?

•  $3.9$  only
  

•  $5$  only

•  $-3.3$  and  $-2.5$  
  

•  $5$  and  $3.9$  

• No Relative Maxima

Multiple Choice

4 represents a(n) _____________.

• Absolute Maximum

• Absolute Minimum

• Relative Maximum

• Relative Minimum

Multiple Choice

The blue dot on this graph represents a(n)...

• Absolute Maximum

• Absolute Minimum

• Relative Maximum

• Relative Minimum

Multiple Choice

What is the Relative Max?

• $36$

• $-6$
  

• $+\infty$
  

• $-\infty$
  

REMEMBER: the ZEROS of a graph are the x-intercepts

Use a graphing calculator OR Desmos (if you don't have a calc) for the next few problems.

Multiple Select

Select the 3 zeros of the polynomial function:

 $3x^3-5x^2+1$  

• -0.40

• 0.55

• 1.12

• 1.52

Multiple Choice

Find the zero(s) of the function.

• (0,1) and (0, -5.5)

• (3, 0) and (-11, 0)

• (1, 0) and (-5.5, 0)

• (0, 3) and (0, -11)

Multiple Choice

Find the zero(s) of the function.

• x = -7.5, 0.33

• x = -0.75, 3

• x = 0.75, -3

• x = -0.33, 7.5

This is the last topic: POSITIVE & NEGATIVE INTERVALS

This graph is positive on the interval (1,2)

because that is when it is above the x-axis!

This graph has 2 positive intervals:

 $\left(-\infty,-1\right)and\ \left(1,\infty\right)$  

Multiple Select

Identify the intervals where the graph is positive. Check all that apply.

•  $\left(-\infty,0\right)$ 

•  $\left(0,4\right)$ 
  

•  $\left(4,\infty\right)$ 

•  $\left(-\infty,2\right)$ 
  

•  $\left(2,\infty\right)$ 

Multiple Select

Identify the intervals where the graph is negative. Check all that apply.

• $\left(-\infty,-1\right)$

• $\left(-1,\infty\right)$
  

•  $\left(-3,1\right)$ 

• $\left(-\infty,-3\right)$
  

• $\left(1.\infty\right)$

Multiple Choice

Identify the interval where the graph is positive.

• $\left(-\infty,-1\right)$

• $\left(-1,\infty\right)$
  

•  $\left(-3,1\right)$ 

• $\left(-\infty,-3\right)$
  

• $\left(1.\infty\right)$

Multiple Choice

The shaded region represents...

• Decreasing Interval

• Increasing Interval

• Constant Interval

• Positive Interval

• Negative Interval

Poll

How well do you understand how to find solutions on the graphing calculator?

• Very well

• Okay

• Not well

Poll

How well do you understand the concepts of max and min?

• Very well

• Okay

• Not well

Poll

How well do you understand the concept of positive and negative intervals?

• Very well

• Okay

• Not well

The next questions will be about our google classroom calendar. Go to the classwork tab and open the calendar document.

Multiple Choice

On what day will we take the Unit 3 test next week?

• Tuesday

• Wednesday

• Thursday

• Friday

Multiple Choice

How many days will we take to review for the midterm?

• 1

• 2

• 3

• 4

Multiple Choice

When we come back to school on January 5th, what math topic will we begin?

• Rational functions

• Logarithmic functions

• Exponential functions

• Calculus functions

LET'S RECAP WHAT YOU'VE LEARNED

1. Graph each side of an equation separately and find the intersection(s).

2. Find maximums and minimums.

3. Find zeros (x-inercepts)

4. Find positive and negative intervals

What to do next:

1. Complete the practice on page 26. I'm attaching it on the next slide for your reference.

2. Check your answers in google classroom by Friday.

3. Check your grades. If you did not take the formative quiz on Friday, 11/20, it will only be open until Friday, 12/5 at 11 pm. It is a google form in google classroom.

THIS IS PAGE 26. DO IT NOW.