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 graph
g(x) = (x + 5)(x -1)
Given 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.
Standard Form
g(x) = x 2 + 4 x - 5
æ -b æ -b ö ö
Vertex ç
,f
è 2a çè 2a ÷ø ÷ø
-b
-4
x=
=
= -2
2a 2(1)
f (-2) = (-2)2 + 4(-2) - 5 = -9
vertex (-2, -9)
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 graph
g(x) = (x + 3)2 (x + 8)
Given 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
f (x) = 2x 3 - 5x 2 -13x +15
f (0) = 2(0)3 - 5(0)2 -13(0)+15
f (0) = 15
•
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 xaxis is positive.
Anything below the xaxis 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 f (x) = f (-x), meaning that it is symmetrical about the yaxis
• A function is odd if f (-x) = - f (x) , 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 1
For 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.
1
f (x) = - (x + 3)(x - 2)(x +1)
2
EXAMPLE CONTINUED
1
f (x) = - (x + 3)(x - 2)(x +1)
2
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
f (x) = - (x + 3)(x - 2)(x +1)
2
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)
These ordered pairs are known as
local maximum and minimum values
Local Minimum Value (-2.12,-2.03) because they are the highest and/or
lowest values in that particular section
of the graph.
Local Maximum Value (0.79,4.10)
EXAMPLE CONTINUED
1
f (x) = - (x + 3)(x - 2)(x +1)
2
4. On what intervals is the function increasing/decreasing?
(0.79,4.10)
Always use the x values to show the
intervals that the function is
increasing/decreasing and
positive/negative
Interval Notation
Decreasing (-∞, -2.12) and (0.79, +∞)
Increasing (-2.12, 0.79)
Inequality Notation
Decreasing x<-2.12 and x>0.79
Negative -2.12<x<0.79
(-2.12,-2.03)
EXAMPLE CONTINUED
1
f (x) = - (x + 3)(x - 2)(x +1)
2
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
1
f (x) = - (x + 3)(x - 2)(x +1)
2
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