Month: April 2018

Pre-Calculus 11 Week 11 – Graphing Inequalities with a Degree of 1

alexr2015

 

We started our inequalities unit this week. If you were told how to graph y=x^2+2, I’m sure you’d be able to graph it. Or y=2x-3? What if I asked you to graph y\leq x+2 How do you graph that?

First, lets refresh what inequality signs mean. The \geq means “greater than or equal to”. For example, 5\geq 2 or 3 \geq 3. While the \leq means “less than or equal to”. For example, 5\leq 7 or 5 \leq 5.

Now that we’ve got that out of the way, now we learn how to graph  an inequality. It’s just like graphing a linear equation, just with an extra step. In y \geq 2x+3, we can see that the inequality is in our slope-intercept form, which you probably learned back in grade 9. Our slope here is 2, and our y-intercept is 3. Make your starting point on the graph your y-intercept, and then draw your slope.

To show our inequality, we highlight the side of the graph, where if we use a point from that highlighted side, the inequality statement will be true.

Okay, okay, that was a mouthful. Just, keep with me, here.

Our last step involves using a “test point”. Now that we’ve graphed our line, we’ve split the graph into two parts. Pick a point that is clearly on one side of the line, say, (-5, -5). Insert the coordinates into your inequality.

y\geq 2x+3

 

-5\geq 2(-5)+3

 

-5\geq -7

 

If the statement is true, then that means for the inequality statement to be true, you have to insert the coordinates for any point that is on that side. If you got a false statement, that means that the other side is correct! Then you just highlight the correct side.

Pre-Calculus 11 Week 10 – Midterm Review!

alexr2015

Our midterm is next week! So for this post, I’ll review one of the units that I had more trouble with. Absolute Values and Radicals. More specifically, rationalising fractions that have binomials as a denominator. Such as…

\frac{3\sqrt{8}+\sqrt2{5}}{\sqrt{2}+\sqrt{20}}

To rationalise a fraction means to make sure that there is no radical in the denominator. In the case that there’s a binomial as the denominator, you still multiply by the conjugate. The conjugate in this case will be a binomial, \frac{\sqrt{2}-\sqrt{20}}{\sqrt{2}-\sqrt{20}}. Multiplying the denominator by it’s conjugate eliminates the “middle term” that it usually creates. This creates a denominator with no radicals! Proof is below… (Excuse my handwriting)

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.