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

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.

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

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

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?

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?

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

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!

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!

Example: What is the vertex for the quadratic equation y = x2 + 4x - 12 ?

Example: Graph the following quadratic equation: y = x2 - 4x - 5

Example: Find the equation of the following quadratic function.

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.

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

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)?

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

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?

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?

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?

b) What is the maximum revenue?