quadratic functions and their characteristics unit 6 quadratic functions math ii
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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 GraphsTRANSCRIPT
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