6.1: exploring quadratic relations...example: solve x 2 - 11x = 0 solve: -24a + 144 = -a2 4m2 + 25 =...
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Math 2201Unit 4: Quadratic Functions
16 Hours
6.1: Exploring Quadratic Relations
Quadratic Relation:☆ A relation that can be written in the standard form y = ax2 + bx + c
Ex: y = 4x2 + 2x + 1
☆ ax2 is the quadratic term☆ bx is the linear term☆ c is the constant term
Parabola:☆ The shape of the graph of any quadratic relation
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Characteristics of a Quadratic Graph (Parabola)
The vertex is where the axis of symmetry meets the parabola. It is the highest or lowest point, called the maximum or minimum.
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The axis of symmetry is a line that divides a parabola into two equal parts that would match exactly if folded over on each other.
☆ The axis of symmetry will always pass through the vertex of the parabola
☆ The x-coordinate of the vertex is used in the equation of the axis of
symmetry
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Identify the following:
a) vertex
b) direction of opening
c) x-intercepts
d) y-intercept
e) line of symmetry
6.2: Properties of Graphs of Quadratic Functions
☆ The value of is the x-coordinate of the vertex, as well as the
equation of the line of symmetry.
☆ The y-coordinate of the vertex can be found by substituting the x-coordinate into the quadratic function
Ex: Find the vertex and axis of symmetry for the parabola. Maximum or minimum?
y = 3x2 + 6x + 2
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The axis of symmetry can also be linked to the x-intercepts of the graph of a quadratic function.
Ex: What are the x-intercepts? How can we determine the axis of symmetry from this information?
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a) Vertex:
b) Axis of Symmetry:
c) x-intercepts:
Sketch the graph y = -x2 + 5x + 4. Consider the vertex, y-intercept, direction of opening, axis of symmetry, and x-intercepts. What is the domain and range of the function?
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6.3: Factored Form of a Quadratic Function
Factored Form:
☆ y = a(x - r)(x - s)
Zero Property:
If a · b = 0 then a = 0, b = 0 or both a and b equal 0
Example: Solve (3x + 5)(x - 3) = 0
Steps for Solving a Quadratic Equation by Factoring
☆ Set the equation equal to 0
☆ Factor the equation (GCF, Box Method)
☆ Set each part equal to 0 and solve
☆ Verify!
Example: Solve x2 - 11x = 0
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Solve: -24a + 144 = -a2 4m2 + 25 = 20m
Determine the zeroes of the following quadratic equation:y = 9x2 - 4
The zeroes of an equation are the x-intercepts of the graph!
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Determine the roots of the following quadratic equation:y = 2x2 + 5x - 3
Determining the Vertex of a Parabola from an Equation
☆ Find the zeroes of the quadratic equation
☆ These zeroes represent the x-intercepts of the graph of the quadratic equation
☆ Average the two zeroes (x-intercepts). This represents the x-coordinate of the vertex of the graph
☆ Substitute the x-coordinate back into the quadratic equation and solve. This will represent the y-coordinate of the vertex.
☆ Write the x and y coordinates as a coordinate pair.
☆ This is the vertex of the parabola!
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Example: What is the vertex for the quadratic equation y = x2 + 4x - 12 ?
Example: Graph the following quadratic equation: y = x2 - 4x - 5
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Example: Find the equation of the following quadratic function.
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Example: Write y = 2(x + 4)(x - 3) in standard form.
Example: Determine the equation of the quadratic function, in factored and standard form with factors (x + 3) and (x - 5) and a y-intercept of -5.
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6.4: Vertex Form of a Quadratic Function
Vertex Form: y = a(x - h)2 + k
☆ If 'a' is positive, the parabola opens up☆ If 'a' is negative, the parabola opens down
☆ The vertex is the point (h, k)☆ The axis of symmetry is x = h
Example: y = 2(x - 1)2 + 3
a) What is the direction of opening?
b) What are the coordinates of the vertex?
c) What is the equation of the axis of symmetry?
Writing an Equation of a Graph in Vertex Form
☆ Use the form y = a(x - h)2 + k
☆ Identify the vertex of the graph and substitute it into the equation for h and k
☆ Identify an additional point on the graph and substitute into the equation for x and y
☆ Solve for a
☆ Write the equation y = a(x - h)2 + k, filling in the a, h, and k values
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Determine the equation of the quadratic function in vertex form
Example: What is the equation of the function with vertex (1, 2) and with a point on the graph passing through (3, 4)?
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Example: Find the equation of the parabola with x-intercepts of (4, 0) and (-8, 0), with a maximum value of 10.
Example: Convert the following equation to Standard Form.
y = 2(x - 3)2 + 5
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Example: A soccer ball is kicked from the ground. After 2 s, the ball reaches its maximum height of 20 m. It lands on the ground at 4 s.
a) Determine the quadratic function that models the height of the kick.
b) What is the domain and range of the function?
c) What was the height of the ball at 1 s? When was the ball at the same height on the way down?
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6.5: Solving Problems Using Quadratic Function Models
1. Determining the maximum height given the quadratic function:
The path of a rocket is given by the equation h = -3t2 + 30 t + 73, where h is the height of the rocket in metres and t is the time in seconds.
a) What is the maximum height of the rocket?
b) At what time does the rocket reach its maximum height?
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2. Area Questions:
A rectangular field, bounded on one side by a lake, is to be fenced on 3 sides by 800 m of fence. What dimensions will produce a maximum area?
3. Revenue Questions:
Labrador Outfitters provides hunting and fishing guides for people outside the province. Last year, there were 1020 guests who each paid $180 per night. Management estimates that for each $1.00 reduction in price, there will be 5 extra customers.
a) At what price would the maximum revenue be reached?
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b) What is the maximum revenue?