Math 2201 Unit 4: Quadratic Functions. 16 Hours

Transcription

1 Math 2201 Unit 4: Quadratic Functions 16 Hours 6.1: Exploring Quadratic Relations Quadratic Relation: A relation that can be written in the standard form y = ax 2 + bx + c Ex: y = 4x 2 + 2x + 1 ax 2 is the quadratic term bx is the linear term c is the constant term Parabola: The shape of the graph of any quadratic relation

2 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.

3 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

4 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 = 3x 2 + 6x + 2

5 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?

6 a) Vertex: b) Axis of Symmetry: c) x-intercepts: Sketch the graph y = -x 2 + 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?

7 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 x 2-11x = 0

8 Solve: -24a = -a 2 4m = 20m Determine the zeroes of the following quadratic equation: y = 9x 2-4 The zeroes of an equation are the x-intercepts of the graph!

9 Determine the roots of the following quadratic equation: y = 2x 2 + 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!

10 Example: What is the vertex for the quadratic equation y = x 2 + 4x - 12? Example: Graph the following quadratic equation: y = x 2-4x - 5

11 Example: Find the equation of the following quadratic function.

12 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.

13 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) 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

14 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)?

15 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

16 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?

17 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 = -3t 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?

18 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?

19 b) What is the maximum revenue?