quadratics

Pre-Calculus 11 Week 9 – Different Forms for Quadratic Equations

alexr2015

Quadratic equations can take many different forms. There are three that we learned in this unit. Standard (Sometimes named Vertex) form, General form, and Factored form.

For this unit, the Standard form is generally (Heh.) the most useful, as it can show the most amount of information. The scale, whether it’s minimum or maximum, left/right/up/down translations, and the location of the vertex. That’s a lot. The Standard form is y = a(x - p)^2 + q. I go into more detail regarding the Standard form in my last week post here.

From the standard form, you can go back into the general form by distributing and then simplifying the equation. Treat it like a normal question and eventually, you’ll reach the General form. (NOTE: You cannot go from the Standard form to Factored form. To get from Standard to Factored, you must go Standard->General->Factored.)

The General form is one of the more familiar forms you’re probably used to prior to this unit. The general form can tell us the scale, whether it’s minimum or maximum, and the y-intercept. a is the scale and whether the parabola opens up or down. c represents the y-intercept. However, the middle term, b is almost useless to us. b is a combination of the parabola translating left/right and up/down. It’s hard for us to do anything with it, so it’s disregarded for this unit. The General form is ax^2 + bx + c, just to refresh your memory.

From the General form, you can go to the Standard form by completing the square. To go to factored form, simply factor the equation.

The Factored form is unique, as it is the only form to tell us our roots, or x-intercepts. It can also tell us our scale, and whether the parabola opens up or down. x1 and x2 represents our roots/x-intercepts. The Factored form is a(x - x1)(x - x2).

i don’t know how to do subscript please forgive me.

Pre-Calculus 11 Week 8 – Graphing Quadratic Functions

alexr2015

How do you graph a quadratic function? It’s the same way you would graph a normal equation. But instead of it looking like a line, it looks like this: 

Like, literally, that’s it. That’s all. Just make a table of values, x and y on either side, plug some x values in, calculate for the y values. That’s all. But how bendy and twisty the line is isn’t random. You can spot certain patterns in quadratic functions. Take y = z(x - p)^2 + q for example.

z represents the scale factor (More on that in a bit), and whether or not the parabola will open up or down. If it opens up, z is positive. If it’s negative, the parabola will open down.

p represents if the parabola moves to the left or to the right of the y-axis. If p is +5, then the parabola’s axis of symmetry, or the middle, will be x = 5. However, this means that in the equation, it’ll appear as y = z(x - 5)^2 + q. Because if we remember, a positive and a negative make a negative, so in y = z(x - 5)^2 + q, it’s actually telling us to translate five units to the right. If our equation was y = z(x + 5)^2 + q (Notice the +), then the parabola would be translated 5 units to the left. p also represents the location of the vertex on the x-axis.

q simply shows us where the vertex lies on the y-axis.

As for z, z is usually 1. If the scale factor is one, the parabola follows a 1, 3, 5 pattern. As in one up, one over. Three up, one over. Five up, one over. And so forth. If the scale factor is 2, then all of that is multiplied by two. Two up, one over. Six up, one over. Ten up, one over. And so forth. Scale factor is 3? Three up, one over. Nine up, one over. Fifteen up, one over. And so forth. So the higher the scale factor is, the thinner the parabola is going to be. The lower, the fatter.

http://amsi.org.au/teacher_modules/Quadratic_Function.html

Pre-Calculus 11 Week 7 – Using the Discriminate to Help Solve Quadratic Equations

alexr2015

This week, we learned how to learn if a quadratic equation had one solution, two solutions, or no solutions using the discriminate. What is the discriminate, you may ask? Look at your quadratic equation, see the expression inside the square root? That’s your discriminant. b^2 - 4ac

If you don’t remember, a represents the first term’s coefficient, %latex b$ represents the second/middle term’s coefficient, and %latex c$ represents the third/last term’s coefficient.

Say your quadratic equation is x^2 + 4x - 7 = 0, then the discriminate would be…

(4)^2 - 4(1)(-7) 16 + 28

The discriminant is 44

The discriminant can tell us whether the answer has two solutions, one solution, or no solutions. If the discriminant is positive, then there are two solutions to your quadratic equation. If the discriminant equals zero, there will be one solution. However, if the discriminant is negative, then your quadratic equation has no solutions.

A quadratic equation graphed, showing two solutions.

What’s a solution? Glad you asked, if you attempt to graph a quadratic equation, your line isn’t a line! It’s this little bendy thing, which is called a parabola. For a quadratic equation to have a solution, a part of it has to be touching the x-axis. So, if the para

bola touches two points on the x-axis, then it has two solutions. However, if the apex of the parabola touches the x-axis, then that means the equation only has one solution. And if the parabola doesn’t touch the x-axis? No solutions.

A quadratic equation graphed, showing no solutions.

A quadratic equation graphed, showing one solution. Notice only the very top, the apex of the parabola touches the x-axis.

Pre-Calculus 11 Week 6 – Solving Quadratic Equations Using the Quadratic Formula

alexr2015

This week, we’ve learned how to solve quadratic equations using the quadratic formula, which is different from simplifying When you attempt to solve a quadratic equation, there will be two possible answers. The quadratic formula looks like this (It’s not that scary, bear with me).

Now, in the quadratic formula, a represents the first term, b represents the middle term, and c represents the third term.

Say we’re given the question 3x^2 - 4x - 1 = 0. Here is what our quadratic equation, and our steps will look like:

You now have two answers for x, because you have a +or- sign.

Pre-Calculus 11 Week 5 – Factoring Quadratic Expressions

alexr2015

This week, we learned how to solve quadratic equations. A quadratic equation looks something like this: x^2 + 3x + 2

x^2 + x + n (n representing a number) Is what you’ll see most of the time. To solve these questions you factor it into two binomials. To check if your answer is correct, distribute the two binomials, and your answer should be the same as your quadratic equation.

The best way to find out what your two binomials are is to think about this: what are two numbers that multiply together to make your n term, but also add together to made your middle term? For the above question, your answer is simple. The only numbers that multiply together to make 2 are 1 and 2. If you add 1 and 2 together, you will get your middle term, 3! What about a harder question?

Our question is x^2 + 7x + 12. I wrote down the multiples that could make 12. Then I find out which ones can add together to make our middle term, 7. The multiples 3 and 4 work! So our answer is (x+3)(x+4)!