1. Lesson 2.1 Quadratic Functions
Essential Question: How do you sketch graphs and write equations of parabolas?

2. Before we start… Which function is considered to be quadratic?
𝑦=3 𝑥 3 −5𝑥+2
𝑓 𝑥 =−7𝑥+9
ℎ 𝑥 = 𝑥 2 −3𝑥+7

3. What is a polynomial function?
Let n be a nonnegative integer and let 𝑎 𝑛 , 𝑎 𝑛−1 , …, 𝑎 2 , 𝑎 1 , 𝑎 0 be real numbers with 𝑎 0 ≠0. The function given by
𝑓 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 2 𝑥 2 + 𝑎 1 𝑥+ 𝑎 0
Is called a polynomial function of x of degree n

4. The Graph of a Quadratic Function
Polynomial functions are classified by degree. For instance, the polynomial function f (x) = a, a  0 has degree 0 and is called a constant function.
The polynomial function f (x) = mx + b, m  0 has degree 1 and is called a linear function.

5. What is a quadratic function?
Let a, b, and c be real numbers with a ≠ 0. The function given by 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐 is called a quadratic function.

6. The Graph of a Quadratic Function
The graph of a quadratic function is a special type of U-shaped curve called a parabola.
All parabolas have an axis of symmetry and a vertex.

7. What is the vertex?
The lowest or highest point on a parabola.

8. What is the axis of symmetry?
The line that divides the parabola into mirror images and passes through the vertex.

9. Basic Characteristics of Quadratic Functions
Graph of 𝑓 𝑥 =𝑎 𝑥 2 , 𝑎>0
Domain: −∞, ∞
Range: 0 , ∞
Intercept: 0, 0
Decreasing on −∞, 0
Increasing on 0, ∞
Even function
Axis of symmetry: 𝑥=0
Relative minimum or vertex: 0, 0

10. Basic Characteristics of Quadratic Functions
Graph of 𝑓 𝑥 =𝑎 𝑥 2 , 𝑎<0
Domain: −∞, ∞
Range: −∞ , 0
Intercept: 0, 0
Increasing on −∞, 0
Decreasing on 0, ∞
Even function
Axis of symmetry: 𝑥=0
Relative maximum or vertex: 0, 0

11. Basic Characteristics of Quadratic Functions
For 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐, when the leading coefficient a is positive, the parabola opens upward; and when the leading coefficient a is negative, the parabola opens downward.

12. Library of Parent Functions: Quadratic Function
Graph of 𝑓 𝑥 = 𝑥 2
Domain: −∞, ∞
Range: 0 , ∞
Intercept: 0, 0
Decreasing on −∞, 0
Increasing on 0, ∞
Even function
Axis of symmetry: 𝑥=0
Relative minimum or vertex: 0, 0

13. Sketch the graph of the function and describe how the graph is related to the graph of f (x) = x2.
a. g (x) = –x2 + 1
b. h (x) = (x + 2)2 – 3

14. Standard Form of a Quadratic Function
𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, 𝑎≠0 Axis is the vertical line 𝑥=ℎ Vertex is the point ℎ, 𝑘 When a > 0, the parabola opens upward, and when a < 0, the parabola opens downward.

15. How do you write a quadratic function in standard form?
By completing the square * Group x-terms * Factor * Complete the square * Regroup * Write in standard form

16. How do you sketch the graph of a quadratic function?
Step 1: Identify the form
Step 2: Identify the vertex
Step 3: Graph the vertex
Step 4: Graph the y-intercept
Step 5: Graph the axis of symmetry
Step 6: Find the mirrored image point for the y-intercept.

17. Write the function 𝑓 𝑥 = 𝑥 2 −10𝑥+25 in standard form, identify the vertex of its graph, and describe the graph relative to the graph of 𝑦= 𝑥 2 .

18. Write the function 𝑓 𝑥 = 2𝑥 2 +8𝑥+7 in standard form, identify the vertex of its graph, and describe the graph relative to the graph of 𝑦= 𝑥 2 .

19. Write the function 𝑓 𝑥 = −𝑥 2 −4𝑥+21 in standard form, identify the vertex of its graph, and describe the graph relative to the graph of 𝑦= 𝑥 2 .

20. How do you identify x-intercepts of a quadratic function?
Factor the function
Solve each factor

21. Find the x-intercepts of the graph 𝑓 𝑥 =− 𝑥 2 +6𝑥−8

22. Find the x-intercepts of the graph 𝑓 𝑥 = 𝑥 2 −2𝑥−15

23. How do you write a quadratic function given a vertex and a point?
Use the form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘
Plug the vertex in for h and k
Plug in the point given to find the value of a
Write your final answer

24. Write the standard form of the equation of the parabola whose vertex is 1, 2 and that passes through the point 3, −6 .

25. Write the standard form of the equation of the parabola whose vertex is −4, 11 and that passes through the point −6, 15 .

26. Minimum and Maximum Values of Quadratic Functions
Consider the function 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐 with vertex − 𝑏 2𝑎 , 𝑓 − 𝑏 2𝑎 .
If 𝑎>0, then f has a minimum at 𝑥=− 𝑏 2𝑎 . The minimum value is 𝑓 − 𝑏 2𝑎 .
If 𝑎<0, then f has a maximum at 𝑥=− 𝑏 2𝑎 . Thmaximum value is 𝑓 − 𝑏 2𝑎 .

27. Minimum and Maximum Values

28. Opening and Minimum / Maximum
If a > 0; Opens up. Vertex is minimum.
If a < 0; Opens down. Vertex is maximum.

29. To Find a Minimum or Maximum Value…
Look at the coefficient of the first term.
If it’s greater than zero, it’s a minimum
If it’s less than zero, it’s a maximum
To find the value, calculate the coordinates of the vertex.

30. A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45° with respect to the ground. The path of the baseball is given by the function 𝑓 𝑥 =− 𝑥 2 +𝑥+3, where 𝑓 𝑥 is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

31. A football is thrown at a point 6 feet above the ground at a velocity of 60 feet per second at an angle of 45° with respect to the ground. The path of the football is given by the function 𝑓 𝑥 = −0.0168𝑥 2 +𝑥+6, where 𝑓 𝑥 is the height of the football (in feet) and x is the horizontal distance from the quarterback (in feet). What is the maximum height reached by the football?

32. How do you sketch graphs and write equations of parabolas?

33. Ticket Out the Door
What is the vertex of the function 𝑓 𝑥 = 𝑥−6 2 +3