vertexgraphlesson

Here is our first lesson for Monday March 2, 2009

//Lesson Plan Template//

Content: (Topic)
 * Mathematics A30; Concept E: Relations and Functions
 * Quadratic Functions and the Vertex-graphing form

Teaching strategy:
 * Team-teaching strategy
 * Also, direct teaching (teacher lecture) and indirect teaching (high levels of student involvement)

Learning Objectives:
 * **E.7** To graph quadratic functions of the standard form f(x) = a(x-p)² + q, by determining the vertex, axis of symmetry, concavity, [maximum or minimum values, domain, range, and zeroes].

Assessment:
 * On going assessment: students will be asked to explain to the class what certain terms mean, explain how to do the class example etc. (more so assessment that involves questions and answers)

Pre-requisite Learning:
 * Knowledge of the Cartesian plane (x, y axis, quadrants)
 * Knowledge of solving for (a) variable(s) in a given equation (i.e. solve for x)
 * Knowledge of the meaning of vertex, concavity (up or down), axis of symmetry, x-intercepts and y-intercepts on a graph

Lesson preparation: Equipment/materials:
 * nothing unique

Advanced preparation:
 * None

__**Presentation**__

Set: (10 min)
 * "Name Game" activity where students will write their names and a paragraph about themselves on paper (to be handed-in)
 * Introduction to the lesson (Today we are going to learn how to graph quadratic functions using the vertex-graphing form...)
 * What is a quadratic function and what is not? (including the Your Turn #2 examples on pg. 84)

Development: (25 min)
 * Introduction of the quadratic function which is written in the form f(x) = a(x-p)² + q
 * Write out an example and determine which numbers represent a, p, and q and explain what each variable represents (i.e. a represents concavity, (p,q) are the coordinates of the vertex, x = p which is the axis of symmetry)
 * Introduce to students how to solve for x-intercepts and y-intercepts using the function
 * Lastly, illustrate the sketching of a graph by determining the critical points (i.e. the vertex, axis of symmetry, x-intercepts and y-intercepts)

Closure: (10 min)
 * Today we learned how to draw the graph a quadratic function using the form f(x) = a(x-p)² + q
 * Tomorrow we are going to learn how to solve for a, p, or q given a quadratic function and a particular point
 * Questions? (for clarification)
 * Homework assignment

Extensions:
 * Students will be given time to work on the assigned homework in class
 * Also, re-visit name game

Classroom Management Strategies:
 * While one partner is teaching, the other can walk amongst the students to make sure they are paying attention and are busy at the work of learning
 * To keep the lesson lively and interesting so that students will be more attentive
 * To be consistent with students and make sure to deal with any problem right away
 * Any questions students may have will be answered one at a time

Adaptive Dimension:
 * The concepts and examples are to be explained by the students as well that way others can understand the math more clearly
 * Students are able to work on the assigned homework in pairs or in groups (that way the stronger math students can work with the weaker math students)

Professional Development Plan (targets):
 * Getting to know the students names
 * Time management

Teacher Notes:

//Set:// You've just finished a unit on linear equations (y=mx+b). Now we need to know how to graph equations that don't have this form. We're going to look at how to solve equations with an x². We call this a quadratic equation and looks like y= f(x) = ax² +bx +c where a, b, and c are real numbers and a cannot be 0. Can anyone say why a cannot equal 0? (because it would be the equation of a line). let's look at some examples and you tell me which are and are not quadratic functions: What we will be working with today is quadratic functions in "Vertex graphing form" y= a(x-p)² +q When you graph it, it forms a parabola.
 * f(x)= 3x-5 (not)
 * f(x)= 7x -x² +3 (yes)
 * y= (x-3)(x+1) (yes)
 * f(x)= -3(x+2)² -8 (yes)
 * y= x² (yes)
 * y= 9x² -x³ (not)

//Development://

When we wish to draw the graph of a quadratic function, the function is written in the form f(x) = a(x-p)² + q where a, p, and q an real numbers and a cannot equal 0. Does anyone know what these variables stand for?

Write out example : y = 2 ( x - 2 ) ² - 4

Looking at the quadratic equation above, what is our a value? a = 2. If we know that a is a positive number, our parabola on the graph will be concave up. If we know that a is a negative number, our parabola on the graph will concave down. What do I mean by concave up/down can anyone tell me? (could show an illustration). Also, if absolute value a = 1, the parabola has a "normal" width. If absolute value a increases beyond 1, the parabola is very narrow. And if absolute value a decreases towards 0, the parabola is very wide.

Next is determining our vertex. What is the vertex? (the lowest or highest point on a parabola) (could show an illustration). For the function, the vertex is always located at the point (p, q). In this particular example, what is p and what is q? p = 2 and q = -4 (make sure you don't confuse the sign of p). So the vertex of the parabola is (2, -4).

Lastly is the axis of symmetry. What is the axis of symmetry? (the vertical line that passes through the vertex cutting the parabola in half) (could show an illustration). Since it is a vertical line and passes through the x-axis, x = p. Hence, in this example x = 2.

Class example: y = 5 ( x - 3 ) ² + 1

i) What are the coordinates of the vertex? (3, 1) ii) Concave up or down? a = 5 (a positive number) so it concaves up iii) What is the axis of symmetry? x = p, p = 3 so x = 3

Now that we know how to read the equation, let's graph it. (graph the question that was just done). One more example: y= -3x² +2 First we need to make it look like our other questions in vertex graphing form. what is a, p, q? y= -3(x-0)² +2 graph it: Further example if needed: -y+4 = -2(x-6)² (make it look like vertex graphing form first, then graph it using steps 1-4). //Closure://
 * 1) vertex (0,2)
 * 2) concavity down
 * 3) x/y-intercepts - Question: How do you know how wide it is? x-intercepts - where the parabola crosses the x-axis (y=0), y-intercepts - where the parabola crosses the y-axis (x=0).
 * 4) Using your axis of symmetry ( x=0), you can draw the parabola.

The assigned questions for the homework are found on pg. 88 and include #1 a, c, d, h #2 b, c, e, f #3 c, h #4 a, c