ANAZYING GRAPHS OF QUADRATICS AND

POLYNOMIAL FUNCTIONS

Overview of this Lesson

• You have learned in the past 2 lessons, how to graph/sketch quadratic functions & polynomial functions.

• In this lesson, we are going to find intercepts, maximum & minimum values, where graphs are increase vs. decreasing, where graphs are positive and negative, discuss end behavior of graphs, symmetry of graphs, and domain and range of these functions.

Warm-up Quadratic Function

Sketch the following graphGiven the equation in intercept form, you can plot the x-intercepts. To find the vertex, the easiest way would be to find the standard form of the equation and the find the vertex.

The “a” value is positive meaning the parabola opens up (lesson 1) and the end behavior (lesson 2) then both end approach

Warm-up Polynomial Function

Sketch the following graphGiven the equation in this form, you can plot the x-intercepts. In lesson 2 you learned how to sketch the graphs of polynomials using the x-intercepts and their end behaviors.

In this lesson you will learn how to•find intercepts when polynomials are not given in intercept form•find the maximum and minimum values •State where the function is increasing and decreasing•State where the function is positive and/or negative•Find the domain and range of the function.

Y-Intercepts: Quadratics and Polynomials

To find y intercepts without a calculator

• You will put in 0 for x and then solve for the y coordinate

• Find the y intercept of the function

• The y intercept of the function is 15, the ordered pair is (0,15).

• To find y intercepts with a calculator

• Go to the PDF named TI-83 Y-Intercept PDF under the Notes section for this lesson

x-Intercepts: Quadratics and Polynomials

To find x intercepts with a calculator

• You will following the steps for finding x-intercepts that you learned in lesson 1 about quadratics. Go to the TI-83 X-Intercepts PDF under the notes section for this lesson.

Maximums and Minimums: Quadratics and Polynomials

To find maximum and minimum points with a calculator

• You will following the steps for finding maximum and minimums points that you learned in lesson 1 about quadratics. Go to the TI-83 Maximum and Minimum PDF under the notes section for this lesson.

Increasing and Decreasing: Quadratics and Polynomials

To determine if a function is increasing or decreasing look at the following.

• A function is increasing if the y values increase while the x-values increase. • A function is decreasing if the y values decrease while the x-values

decrease.

Positive and Negative:Quadratics and Polynomials

To determine if a function is positive or negative look at the following.

• A function is positive if the y values of the points on the graph are positive. • A function is negative if the y values of the points on the graph are

negative.

Anything above the x-axis is positive.

Anything below the x-axis is negative.

End Behavior:Quadratics and Polynomials

To determine a functions end behavior recall Lesson 2 notes.

Even and ODD:Quadratics and Polynomials

To determine if a function is even or odd look at the following.

• A function is even if , meaning that it is symmetrical about the y-axis

• A function is odd if , meaning that is it symmetrical about the origin.

Domain and Range:Quadratics and Polynomials

To determine the domain and range of a function you have to think about the x values (domain) and y values (range) of the function.

There are 2 ways to express the domain and range

Option 1- Inequality Notation (using <, >, ≤, or ≥)

Option2 – Interval Notation (using (, ), [, or ] ) Using ( or ) means that the number isn’t included and using a [ or ] means that the number is included.

Look at the examples on the next slide.

Domain and Range:Quadratics and Polynomials

Example 1For the following function please answer the following questions

1.What is the y-intercept? (state your answer as an ordered pair)

2.What are/is the x-intercepts? (state your answer/s as an ordered pair)

3.What are the maximum and/or minimum values (state your answer/s as an ordered pair)

4.On what intervals is the function increasing/decreasing?

5.On what intervals is the function positive/negative?

6.Does the function of even, odd, or no symmetry?

7.What is the domain and range? (state your answers in inequality and interval notation)

Remember to use the TI83 PDFs to help you.

Example Continued

You know how to sketch the graph from lesson 2. You know the end behaviors of the model because the degree will be off and the leading coefficient will be negative so your end behaviors will go to positive infinity as your x-values decrease and negative infinity as your x values increase. Now lets find the rest.

Example continued

1. What is the y-intercept? (state your answer as an ordered pair)

(0,3)

2. What are/is the x-intercepts? (state your answer/s as an ordered pair)

(-3,0) (2,0) (-1,0)

3. What are the maximum and/or minimum values (state your answer/s as an ordered pair)

Local Maximum Value (0.79,4.10)

Local Minimum Value (-2.12,-2.03)

These ordered pairs are known as local maximum and minimum values because they are the highest and/or lowest values in that particular section of the graph.

Example continued

4. On what intervals is the function increasing/decreasing?

(0.79,4.10)

(-2.12,-2.03)

Always use the x values to show the intervals that the function is increasing/decreasing and positive/negative

Interval NotationDecreasing (-∞, -2.12) and (0.79, +∞)

Increasing (-2.12, 0.79)

Inequality NotationDecreasing x<-2.12 and x>0.79

Negative -2.12<x<0.79

Example continued

5. On what intervals is the function positive/negative?

Interval Notation

Positive (-∞, -3) and (-1, 2)

Negative (-3,-1) and (2, +∞)

Inequality Notation

Positive x<-3 and -1<x<2

Negative -3<x<-1 and x>2

6. Does the function of even, odd, or no symmetry?

No symmetry (you can’t reflect over the y-axis and the function be on top of itself AND you can’t reflect it over the y-axis and the x-axis to have the function be on top of itself)

Example Continued

7. What is the domain and range? (state your answers in inequality and interval notation)

Interval Notation

Domain (-∞, +∞)

Range (-∞, +∞)

Inequality Notation

Domain: All Real Numbers

Range: All Real Numbers