quadratic functions and their characteristics unit 6 quadratic functions math ii

Quadratic Functions and their Characteristics

Unit 6 Quadratic FunctionsMath II

Mini Lesson: Domain and RangeRecall 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

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

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

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Write the domain and range in interval notation.

Domain: Range:

Introduction to Domain and Range

Domain and Range worksheet!

Find the Domain and Range of each graph.

Whoever can finish 1st, 2nd, and 3rd with all questions correct will get a piece of candy!

Introduction to Quadratic Functions

• A quadratic function always has a degree of 2.– This means there will always be an x2 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 = ax2 + bx + cWhere a, b, and c can be real numbers with a ≠ 0.

Introduction to Quadratic Functions

Vertex

Axis of Symmetry

Domain: (-∞,∞∞)

Range: [0, ∞)

Most Basic Quadratic Function:

y = x2

Introduction to Quadratic Functions

X-InterceptsRootsZeroesSolutions

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.

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?:

Back to Standard Form!

y = ax2 + bx + 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.

€

x =−b2a

Example: Put each in Standard Form. Determine whether each function is a quadratic.

1. f(x) = (-5x – 4)(-5x – 4)

2. y = 3(x – 1) + 3

3. y = x2 + 24 – 11x – x2

4. f(x) = 3x(x + 1) – x

5. y = 2(x + 2)2 – 2x2

Example: Determine whether the vertex will be a maximum or minimum. Find the Axis of Symmetry and Vertex of each.

1. y = x2 – 4x + 7

2. y = -3x2 + 6x - 9

3. y = 2x2 – 8x + 1

4. y = -x2 – 8x – 15

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.

Example: Find a quadratic model for each set of values.

1. (1, -2), (2, -2), (3, -4)

2. x -1 1 3f(x) -1 3 8

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 BallTime Height 0 s 46 ft

1 s 63 ft

2 s 48 ft

3 s 1 ft