Name: ____________________________
Quadratics, Cubics and Quartics, oh my!
Use Desmos software to graph each of the following functions. Sketch the general shape (quick rough
sketch – doesn’t need to be perfect).
1. Quadratics (2nd degree)
a) 𝑓(𝑥) = 𝑥 2
b) 𝑓(𝑥) = −𝑥 2 + 4
2. Cubics (3rd degree)
a) 𝑓(𝑥) = 𝑥 3
b) 𝑓(𝑥) = 𝑥 3 + 3𝑥^2
c) 𝑓(𝑥) = −2𝑥 3 − 𝑥 2 + 𝑥
3. Quartics (4th degree)
a) 𝑓(𝑥) = 𝑥 4
b) 𝑓(𝑥) = 𝑥 4 − 4𝑥 2 + 1
c) 𝑓(𝑥) = −𝑥 4 + 𝑥 3 + 𝑥 2 + 1
b) 𝑓(𝑥) = −𝑥 5 + 3𝑥 3 − 2𝑥
c) _____________________________
Make up your own and graph it
4. 5th degree
a) 𝑓(𝑥) = 𝑥 5
Name: ____________________________
5. Do the two ends of quadratic (2nd degree) functions go in the same direction or opposite
directions? (circle one)
Same
Opposite
6. Cubics (3rd degree)?
Same
Opposite
7. 4th degree?
Same
Opposite
8. 5th degree?
Same
Opposite
9. If the degree (the largest exponent in the function) is an odd number, do the ends go in the
same or opposite directions?
Same
Opposite
10. If the degree (the largest exponent in the function) is an even number, do the ends go in the
same or opposite directions?
Same
Opposite
Graph the following function on the computer: 𝑓(𝑥) = (𝑥 − 2)(𝑥 + 1).
11. What kind of function is this? (choose from quadratic, cubic, etc.)
12. At what x values does the graph cross the x axis?
13. Because the x axis is where the function equals 0, we call these points “zeros.” You can find
them by solving the equation 0 = (𝑥 − 2)(𝑥 + 1). What are the two solutions to this equation?
Now look at the function 𝑓(𝑥) = (𝑥 − 3)(𝑥 + 5)(𝑥 + 2). Don’t graph it yet! First answer the questions:
14. What kind of function is this? (choose from quadratic, cubic, etc.)
15. Do the ends go in the same or opposite directions?
16. What are the zeros of this function?
17. Now graph it on the computer and see if it goes through the x axis for the numbers you wrote
down for question 16. Sketch the graph here.
Finally, graph the following function: 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 2)(𝑥 + 1)(𝑥 + 3).
18. What kind of function is this? (choose from quadratic, cubic, etc.)
19. Do the ends go in the same or opposite directions?
20. What are the zeros of this function?
21. For each zero, say whether the graph “crosses through” or “bounces off of” the x axis.