1. Quadratic Functions and their Characteristics Unit 6 Quadratic Functions Math II

2. Mini Lesson: Domain and Range Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs. Interval Notation is the way to represent the domain and range of a function as an interval pair of numbers. – Examples: [-2, 3], (0, 2], (-∞, ∞) – The numbers are the end points of the interval – Use parenthesis if the endpoint is NOT included – Use brackets if the endpoint IS included – ∞ (Infinity) : Use this if numbers go on forever in the positive direction – -∞ (Negative Infinity) : Use this if numbers go on forever in the negative direction

3. Find the domain and range of the function graphed to the right. Use interval notation. Domain: [ -3, 4 ] Range: [ -4, 2 ] x y Introduction to Domain and Range Domain and Range from Graphs

4. Introduction to Domain and Range Find the domain and range of the function graphed to the right. Use interval notation. x y Domain: ( - ∞, ∞ ) Range: [ -2, ∞ ) Domain and Range from Graphs

5. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

6. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

7. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

8. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

9. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

10. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

11. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

12. Introduction to Domain and Range Write the domain and range in interval notation. Domain: Range:

13. Introduction to Domain and Range Domain and Range worksheet! Find the Domain and Range of each graph. Whoever can finish 1 st, 2 nd, and 3 rd with all questions correct will get a piece of candy!

14. Introduction to Quadratic Functions A quadratic function always has a degree of 2. – This means there will always be an x 2 in the equation and never any higher power of x. The shape of a quadratic function’s graph is called a parabola. -It looks like a “U” The Standard Form of a Quadratic is y = a x 2 + b x + c Where a, b, and c can be real numbers with a ≠ 0.

15. Introduction to Quadratic Functions Vertex Axis of Symmetry Domain: (-∞,∞∞) Range: [0, ∞) Most Basic Quadratic Function: y = x 2

16. Introduction to Quadratic Functions X-Intercepts Roots Zeroes Solutions

17. Quadratic vocabulary - Axis of Symmetry – the line that divides a parabola into 2 parts that are mirror images. The axis of symmetry is always a vertical line defined by the x-coordinate of the vertex. (ex: x = 0) - Vertex – the point at where the parabola intersects that axis of symmetry. The y-value of the parabola represents the maximum or minimum value of the function. - X-Intercepts, Zeroes, Roots, Solutions – All of these terms mean the same thing and refer to where the parabola crosses the x-axis. When asked to solve a quadratic, this is what we are looking for! - Minimum – If the graph opens up (smiles) and the vertex is the lowest point on the graph, then the vertex is a minimum. - Maximum - If the graph open down (frowns) and the vertex is the highest point on the graph, then the vertex is a maximum.

18. Example: Find the vertex, axis of symmetry, zeroes, and domain and range of the quadratic function. Determine if the vertex is a minimum or a maximum. Vertex: Axis of Symmetry: Zeroes: Domain: Range: Min or Max?: Vertex: Axis of Symmetry: Zeroes: Domain: Range: Min or Max?:

19. Back to Standard Form ! y = a x 2 + b x + c -If a > 0, then vertex will be a minimum. -If a < 0, then vertex will be a maximum. -If in standard form, use the formula to find axis of symmetry. – This will also be the x coordinate of the vertex, substitute that into the original equation to find the y coordinate.

20. Example: Put each in Standard Form. Determine whether each function is a quadratic. 1.f(x) = (-5x – 4)(-5x – 4) 1.y = 3(x – 1) + 3 1.y = x 2 + 24 – 11x – x 2 1.f(x) = 3x(x + 1) – x 1.y = 2(x + 2) 2 – 2x 2

21. Example: Determine whether the vertex will be a maximum or minimum. Find the Axis of Symmetry and Vertex of each. 1.y = x 2 – 4x + 7 1.y = -3x 2 + 6x - 9 1.y = 2x 2 – 8x + 1 1.y = -x 2 – 8x – 15

22. Finding Quadratic Models! (in your calculator) Find a quadratic model for a set of values. - Step 1 : Enter the data into the calculator (STAT -> Edit) (x’s in L1, y’s in L2) - Step 2 : Calculate the Quadratic Regression model by hitting STAT again  Calc, then 5: QuadReg - Step 3 : Substitute the given a, b, and c values into the standard form.

23. Example: Find a quadratic model for each set of values. 1.(1, -2), (2, -2), (3, -4) 2. x 13 f(x) 38

24. Quadratic Model Application A man throws a ball off the top of a building. The table shows the height of the ball at different times. a.Find a quadratic model for the data. b.Use the model to estimate the height of the ball at 2.5 seconds. Height of a Ball TimeHeight 0 s46 ft 1 s63 ft 2 s48 ft 3 s1 ft