Quadratics Essentials MPM2D

Okay, so we’ve been studying Quadratics for a LONG time and by now there are many things that you should be able to do with them. Let’s get this all straight so that we can apply our new knowledge to different situations.

The SKILLS you need:

The BASIC forms of the quadratic:

a)  STANDARD FORM:

What we can tell from the SF: The y-intercept. It’s RIGHT there … the letter c How do I know that, you ask? Well, when x = 0 everything on the rhs becomes zero except the c, and we know that the y-intercept is on the y-axis and ALL coordinates of points on this axis have x = 0 for their x value.

b)  FACTORED FORM:

What can we tell from the FF: The Zeros, or x-intercepts. All you have to do to find them is set EACH set of brackets equal to zero and solve for x. Why do you do that, you ask? Well, first of all we are trying to find out where y = 0, because that would mean that we are on the x-axis (all coordinates on the x-axis have y = 0), so we are trying to find what will make the rhs of the equation equal to zero, and we know that everything on the rhs is multiplied together and that when we multiply anything by zero our answer is always 0.

c)  VERTEX FORM:

What can we tell from the VF: SO much information! Because it is called the VERTEX form, obviously the vertex must be obvious, and yes, it is! (h ,k) is the vertex!

Remember that if the equation is something like: , the vertex would be

(3, -4) Notice that the x-coordinate has the opposite sign and that we transformed the parabola 3 units to the right!. The vertex form also allows us to state ALL of the transformations that have been performed on the “parent” function of the most basic of all parabolas .

Working with Polynomials

In this unit we learned how to expand and how to factor, which are opposite operations!

You should, by now, be able to expand something like this:

Remember the little saying? “SQUARE, TWICE THE PRODUCT, SQUARE”

Try it below, say it out loud! This is what you should do every time you square a binomial.

=

You should also be able to expand two binomials … make the little rainbows!

Also, we did LOTS and LOTS of factoring and instead of re-writing all the rules again for you, I encourage you to read over all the methods in the handout you received called “Factoring Polynomials – a Review” which I have posted on the website if you have lost it!

TRANSFORMATIONS

To easily identify the transformations, we put the quadratic in vertex form. Once we have done that we can read them from left to right! Remember that you might not have ALL the transformations happening with the quadratic you are working with!

1. a - tells you the vertical stretch or compression of the parabola. If the number is between 0 and 1 (i.e. a fraction less than 1), then there is a compression. If the number is greater than 1 (it does NOT matter that it is negative), it is a vertical stretch by a factor of that number.

- if the “a” value has a negative sign, it means that the graph has been reflected about the x-axis.

2. h - tells you the horizontal shift of the parabola. The changes to x are all INSIDE brackets, because x’s are WEIRD. So if it says (x – 2) there is a horizontal shift RIGHT 2 units.

3. k - tells us the vertical shift of the parabola. It can be up “k” units if k is positive or down “k” units if k is negative.

Finding Equations of Quadratics

Factored form You are always given the zeros and one of the following 3 other scenarios / Vertex form
You are always given the vertex and one of the following 2 other scenarios
1. y-intercept ( 0 , y-int value) / 1. y-intercept ( 0 , y-int value)
2. another point ( x, y ) / 2. another point ( x, y )
3. Max or min (? , y)

In each of the above examples you are trying to find the “a” value, because, as you know, the “a” tells you the shape of the parabola, whether it is vertically compressed, stretched or reflected. All you have to do is substitute in the given values for s, t, x, y (for the factored form) OR h, k, x, y (for the vertex form) and solve for “a”.

For scenario 1, you just have to remember that the x value is 0, for the 2nd scenario, just plug everything in and for the third scenario … well, you will need to do a bit more work to find the x value! For scenario 3 you will need to add the zeros together and divide by 2 to find the axis of symmetry, which gives you the x-coordinate of the vertex (max or min value of the parabola).”

QUADRATIC FORMULA

This formula comes in very handy when you can’t factor (whether it is NOT a factorable equation or you simply can’t remember how to factor)! It will work! All you have to do is make sure that your quadratic is written in STANDARD form first, so that the a, b, and c values have the right signs and then all you have to do is plug them into the magic formula and solve for x!

Remember that the sign means that there will be 2 solutions for your x-intercepts (provided that the parabola crosses the x-axis).

How will you know if your parabola doesn’t cross the x-axis? The value under the radical sign () will be negative. We can NOT take the square root of a negative number (because any number squared will be positive!).

It simply means that the parabola has a vertex above the x-axis and is concave up (the “a” value is positive), OR it has a vertex

below the x-axis and it is concave down (the “a” value is negative)

like you see to the right. Neither graph will cross the x-axis.

FINDING THE VERTEX

You are often asked to find the vertex in quadratic word problems. The question won’t say “FIND THE VERTEX” … that would be too obvious! Instead it will ask you to find a minimum value, a maximum value, the optimal value or how high the ball/Frisbee/rocket etc went or how low something else went. But that’s okay, because you know at least 4 ways to find the vertex:

1.  Find the zeros, add them up, divide by two and you have the x-coordinate of the vertex. Plug that value into the original equation and voila … the y-coordinate.

2.  Complete the square! Not such a hard task, but one that shows that you learned your grade 10 lesson well! See the handout for specifics.

3.  Using will give you the x-coordinate of the vertex and again you just plug that in to find the y-coordinate.

4.  Partial Factoring: See the handout that I gave you for the method.