Quadratic Relationships Website
BY: MITHULAAN SRISKANTHARAJAH
TABLE OF CONTENTS
- INTRO TO QUADRATICS
- USING TABLES OF VALUES, TECHNOLOGY AND FIRST DIFFERENCES TO ANALYZE QUADRATICS
- INVESTIGATING VERTEX FORM
- GRAPHING VERTEX FORM
- FINDING EQUATION AND ANALYZING VERTEX FORM
- FACTORED FORM
- MULTIPLYING BINOMIALS
- COMMON FACTORING AND SIMPLE GROUPING
- FACTORING TRINOMIALS WITH LEADING 1
- FACTORING TRINOMIALS WITHOUT LEADING 1
- FACTORING SPECIAL TRINOMIALS
- COMPLETING THE SQUARE
- QUADRATIC FORMULA
- WORD PROBLEMS
Introduction to Quadratics
This chapter deals with equations involving quadratic polynomials, i.e. polynomials of degree two. Quadratic equations are equations of the form y = ax2 + bx c or y =a(x - h)2 + k . You will then use this equation to create a graph to represent the equation. The shape of the graph of a quadratic equation is a parabola.Some important features of the parabola you should know include the vertex, zero(s), axis of symmetry, y-intercept and optimal value (refer to image: Introducing the Parabola to clarify).Now here is a quick list of everything you could ever want to know about parabolas:
- Parabolas can open up and down
- The zero of a parabola is where the graph crosses the x-axis
- "Zeros" can also be called "x-intercepts" or "roots"
- The axis of symmetry divides the parabola into two equal halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is also the minimum or maximum value
- The optimal value is the value of the y co-ordinate of the vertex
- The y-intercept of a parabola is where the graph crosses the y-axis
Using Tables of Values, Technology and First Differences to Analyze Quadratics
One way to analyze quadratics to make a table of values that includes columns for the x-axis, the y-axis, the first difference and second difference. Just to recap a bit of of Linear relationships they have the same First difference throughout the table. So since we're working with Quadratics we will use the SECOND differences and make sure they're all the same to confirm it's a quadratic relationship.
Investigating Vertex Form
So far we have mainly been expressing the equation in standard form (y-ax2 + bx) but there is another form called vertex form that is expressed like[ y =a(x - h)2 + k]. For more clarification please refer to the Standard and Vertex form image below. The vertex form can tell us that :
- If it is positive then the parabola opens upwards like a regular "U"
- If it is negative, then the graph opens downwards like an upside down "U"
- The "h" value moves the vertex left or right
- The "k" value moves the vertex up or down
- The "a" value stretches the parabola
Graphing Vertex Form
As we all know the base of vertex form is [ y =a(x - h)2 + k] and the h value moves the vertex left or right, the k value moves the vertex up or down and the a value stretches the parabola.Also the last bit of recap the "stretch pattern"is how a regular parabola (with no value for "a") moves from the vertex and up; it moves over one ,up one then over two, up four. So as usual you start off by making a table of values and fill that out. Then you can just look at an equation in vertex form eg: y=(x-1)2 - 4 and figure out the vertex by looking at the h and k values. So in this example it would be (1, -4) and since there is no value for a the stretch pattern applies here. However if there was a value for the same equation like y=2(x-1)2 - 4 the vertex would stay the same however instead of going the stretch pattern we multiply it by 2 (in other equations w.e the multiplier is) so instead of going over one ,up one then over two, up four we'd go over one ,up two then over two, up eight (differences are in bold).
- The vertex determines where we start graphing the equation
- The "a" value tells us if the step pattern stays the same or multiplies by the value
Finding Equation and Analyzing Vertex Form
Sometimes you will be given just the vertex of a a parabola and will be asked to find the rest. For example lets take a vertex of (-2, 8). If you remember from the previous post you learned that the vertex will move 2 units to the left and 8 units up. Now if we were to insert this into the base formula it would look like, y= ? (x+2)2+8. If you look at the equation the "a" value is missing unfortunately you cant do anything until you are given some more information like a x-intercept. For example we are given a x-intercept of this parabola (2,0) we sub in this into the equation of y= ? (x+2)2+8.This will look like 0=a(2+2)2+8, we will then follow BEDMASS and should end up with an answer of a = -1/2
Factored form is expressed as y=a(x-r)(x-s) for example y=0.5(x-3)(x-9) It is basically showing the factors that make up a number. When you want to graph an equation in factored form you will first have to find the "zeros" or x-intercepts. To do this you have to set y to 0 then solve. After solving you should get -3 and 9 as zeros. Then to find the axis of symmetry you will use the Mid-Point formula that will look like this x=-3+9/2. After solving you should get x=3
When multiplying binomials you are basically applying the distributive property twice. The following video will help clarify and confusion.
Introducing the Parabola
Clearly labelled important features on a parabola.
Standard and Vertex form
Here is a visual representation of the equation of a parabola being expressed in standard and vertex form.
Common Factors and Simple Grouping
Common Factoring is when you find the Greatest Common Factor (GCF) between polynomials to simplify them. After you do this you put what ever the GCF outside the brackets and whatever is left from the polynomials inside the brackets.
Factoring Trinomials With Leading 1
Factoring Trinomials with leading 1 ( Simple Trinomials )
Factoring Trinomials Without Leading 1 ( Complex Trinomials)
Factoring Special Trinomials
Special trinomials will come in the formula : a^2 - b^2 . And when you solve you should get (a + b)(a - b). This is also called difference of squares. For this to work both a and b must be perfect squares. Here are some examples.
Completing the Square
Completing the Square is used for getting from standard form to vertex form.
Throughout this unit I have learned many new and different concepts about math. I think the word problems were the best part as I was able to connect the math I do in the classroom to the real world. Although some concepts were difficult they were taught in a good way. I did well on quizzes which proves I understand the concepts now.