Module 7 Part 2: Quadratic Equations

Quadratic Parent Function *Parabola: *Axis of Symmetry: *Vertex

CCSS Instructional Focus Interpret Key Features from

Expressions (F.IF.8) I can find the vertex, axis of symmetry, direction of opening, and zeros/roots in a

quadratic function in context of a situation using a variety of methods.

Quadratic Function

Standard form of a Quadratic Function

What is the vertex of each graph? Is it a minimum or a maximum? 1. 2. 3. Why do you think 𝑎 ≠ 0? 4. What is the order, from widest to narrowest, of the graphs of the functions?

𝑓(𝑥) = −𝑥2 , 𝑓(𝑥) = 3𝑥2 , and 𝑓(𝑥) = −(1/3)𝑥2 ?

All About “a” a>0: a<0:

All About “a” -- Comparing width |a|>1: 0<|a|<1

5. Does changing the value of “c” change the axis of symmetry?

All About “c” when b=0 𝒚 = 𝒙𝟐 + 𝟐 𝒚 = 𝒙𝟐 − 𝟐 What does the “c” value tell us about the graph when b = 0?

6. An acorn drops from a tree branch 70 ft above the ground. How can we graph the height h of the acorn (in feet) after t seconds? At what time does the acorn hit the ground?

Falling Object Model When an object falls, its speed continues to increase, so the height from the ground decreases at a faster and faster rate. Ignoring air resistance, you can model the object's height with the function ℎ = −16𝑡2 + 𝑐.

The Axis of Symmetry *The axis of symmetry is an invisible vertical line. This vertical line divides a parabola into two equal halves. The line of symmetry ALWAYS runs through the vertex of the parabola. How do we represent this line as an equation? Finding the Axis of Symmetry

Y-intercept of Quadratic Functions

7. What is the graph of the function 𝑦 = −𝑥2 + 4𝑥 − 2? 8. During halftime of a basketball game, a slingshot launches T-shirts at the crowd. A T-shirt is launched with an initial upward velocity of 64 ft/s from 5 ft above court level. The T-shirt is caught 35 ft above the court. How long will it take the T-shirt to reach its maximum height? How far above court level will it be? What is the range of the function that models the height of the T-shirt over time?

Vertical Motion Model When an object is projected into the air given an initial upward velocity, v, continues with no additional force acting on it, the formula ℎ = −16𝑡2 + 𝑣𝑡 + 𝑐 gives its approximate height above the ground.

9. What is the graph of the function 𝑦 = 3𝑥2 + 6𝑥 + 2? 10. What is the graph of the function 𝑓(𝑥) = (𝑥 − 1)2?

11. What is the graph of the function 𝑓(𝑥) = 2𝑥2 + 2𝑥 + 1? 12. What is the graph of the function 𝑓(𝑥) = −2𝑥2 + 12𝑥 − 2?

13. What is the graph of the function 𝑓(𝑥) =3

2𝑥2 + 6𝑥 + 2?

14. What is the graph of the function 𝑓(𝑥) = 𝑥2 + 4𝑥?

15. What is the graph of the function 𝑓(𝑥) = −3

4𝑥2 + 2𝑥 + 3?

16. What is the graph of the function 𝑦 = 2𝑥2 + 3?

17. What is the graph of the function 𝑦 = −2𝑥2 − 8𝑥? 18. What is the graph of the function 𝑓(𝑥) = (𝑥 − 2)2 + 1?

Vertex form of a Quadratic Function

19. What is the graph of the function 𝑓(𝑥) = 2(𝑥 + 2) 2 − 1? 20. Graph 𝑓(𝑥) = (𝑥 + 3) 2 − 2. 21. Graph 𝑦 = (𝑥 − 4) 2 − 5.

22. Graph 𝑦 = − 1

2(𝑥 + 4)2 + 6. 23. Graph 𝑓(𝑥) =

1

3(𝑥 + 3)2 − 1

24. Graph 𝑦 = 2(𝑥 − 5)2.

25. Write the quadratic equation of the graph in vertex form, then change your equation into standard form.

26. Write the quadratic equation of the graph in standard form.