The document provides polynomial functions along with their factored forms and instructions for graphing them using GeoGebra. It discusses characteristics of the polynomials such as leading coefficients, degree types, and the number of turning points, as well as describing the end behaviors of their graphs. Each function is analyzed to determine its behavior in terms of rising or falling to the left and right.

1. 1. y =𝟐𝐱 𝟑 − 𝟕𝐱 𝟐 − 𝟕𝐱 + 𝟏𝟐
2. y= −𝒙 𝟓 + 𝟓𝒙 𝟑 − 𝟒𝒙
1.y =𝐱 𝟒 − 𝟕𝐱 𝟐 + 𝟔𝐱
2. y= −𝒙 𝟒 + 𝟐𝒙 𝟑 − 𝟏𝟒𝒙 – 24

2. Factor the following
expressions!

3. 1. y =𝟐𝐱 𝟑 − 𝟕𝐱 𝟐 − 𝟕𝐱 + 𝟏𝟐
y = (2x+3)(𝒙 − 𝟏)(𝒙 − 𝟒)
2. y= −𝒙 𝟓
+ 𝟓𝒙 𝟑
− 𝟒𝒙
y = −𝒙(𝒙 + 𝟏)(𝒙 − 𝟏)(𝒙 + 𝟐)(𝒙 − 𝟐)

4. 1. y =𝐱 𝟒 − 𝟕𝐱 𝟐 + 𝟔𝐱
y = x(x+3)(𝒙 − 𝟏)(𝒙 − 2)
2. y= −𝒙 𝟒 + 𝟐𝒙 𝟑 − 𝟏𝟒𝒙 – 24
y = −(𝒙 + 𝟑)(𝒙 + 𝟏)(𝒙 − 𝟐)(𝒙 − 𝟒)

6. YOUR ACTIVITY!
Graph of the following functions using
Geogebra.
1. y = 2x3 – 7x2 – 7x + 12
2. y = –x5 + 5x3 – 4x
3. y = x4 – 7x2 + 6x
4. y = – x4 + 2x3 + 13x2 – 14x – 24

8. y= −𝒙 𝟓
+ 𝟓𝒙 𝟑
− 𝟒𝒙

9. y =𝒙 𝟒
− 𝟕𝒙 𝟐
+ 𝟔𝒙

10. y= −𝒙 𝟒 + 𝟐𝒙 𝟑 − 𝟏𝟒𝒙 – 24

12. • Is the leading coefficient a positive or a
negative number?
• Is the polynomial of even degree or odd
degree?
y = (2x+3)(𝒙 − 𝟏)(𝒙 − 𝟒)
Positive Leading Coefficient
Odd Degree

13. • How many turning points do the graph
has?
• Observe the end behaviors of the graph
on both sides. Is it rising or falling to the
left or to the right?
y = (2x+3)(𝒙 − 𝟏)(𝒙 − 𝟒)
Two(2) turning points
The graph is falling to the left and rising to
the right.

14. • Is the leading coefficient a positive or a
negative number?
• Is the polynomial of even degree or odd
degree?
Negative Leading Coefficient
Odd Degree
y= −𝒙 𝟓
+ 𝟓𝒙 𝟑
− 𝟒𝒙
y = −𝒙 𝒙 + 𝟏 𝒙 − 𝟏
(𝒙 + 𝟐)(𝒙 − 𝟐)

15. • How many turning points do the graph
has?
• Observe the end behaviors of the graph
on both sides. Is it rising or falling to the
left or to the right?
Four(4) turning points
The graph is rising to the left and falling to
the right
y= −𝒙 𝟓
+ 𝟓𝒙 𝟑
− 𝟒𝒙
y = −𝒙 𝒙 + 𝟏 𝒙 − 𝟏
(𝒙 + 𝟐)(𝒙 − 𝟐)

16. • Is the leading coefficient a positive or a
negative number?
• Is the polynomial of even degree or odd
degree?
Positive Leading Coefficient
Even Degree
y =𝐱 𝟒
− 𝟕𝐱 𝟐
+ 𝟔𝐱
y = x(x+3)(𝒙 − 𝟏)(𝒙 − 2)

17. • How many turning points do the graph
has?
• Observe the end behaviors of the graph
on both sides. Is it rising or falling to the
left or to the right?
Three(3) turning points
The graph is rising to the left and to the
right
y =𝐱 𝟒
− 𝟕𝐱 𝟐
+ 𝟔𝐱
y = x(x+3)(𝒙 − 𝟏)(𝒙 − 2)

18. • Is the leading coefficient a positive or a
negative number?
• Is the polynomial of even degree or odd
degree?
Negative Leading Coefficient
Even Degree
y = −𝐱 𝟒 + 𝟐𝐱 𝟑 − 𝟏𝟒𝐱 – 24
y = − 𝐱 + 𝟑 𝐱 + 𝟏
(𝐱 − 𝟐)(𝐱 − 𝟒)

19. • How many turning points do the graph
has?
• Observe the end behaviors of the graph
on both sides. Is it rising or falling to the
left or to the right?
Three(3) turning points
The graph is falling to the left and to the
right
y = −𝐱 𝟒 + 𝟐𝐱 𝟑 − 𝟏𝟒𝐱 – 24
y = − 𝐱 + 𝟑 𝐱 + 𝟏
(𝐱 − 𝟐)(𝐱 − 𝟒)

20. FRANZ BROQUEZA
DEMO TEACHER