Week 13/Graphing Reciprocal Functions

Reciprocal Functions

Reciprocal Functions is taking a function like y= 2x^2+12x-20 and making it y=1/ (2x^2+12x-20) is the functions reciprical.

When finding y= 2x^2+12x-20 and its reciprical y=1/ (2x^2+12x-20), you want to graph the original function first all the time. After you have graphed the original function, you have to find its vertical asymptote which is where the graph touches the x axis and where the reciprocal will never cross over. It acts like a dotted line on going vertically upwords to help you graph the reciprical. There is also a horizontal asymptote which is for now always on the a axis, it acts simular to the vertical asymptote except it sits vertically on the x axis making sure that your reciprical never touches it.

Once you have graph the parabola, now that you have found that there is no x intercept in the equation, so there is no vertical asymptote. This is what you would call a pimple graph, one that does not have a vertical asymptote but does however still have a horizontal asymptote. To find out where it recipricates you have to find the vertex on the graph.

(3,-2) now that the vertex is found, you can recipricate the vertex, but only the -2 or the y, then you get (3,-1/2) and then you draw the bump that carries on through the x axis and down towards the new vertex for the reciprocal function.

Week 12/ Absolute value functions

Absolute value functions

I’ll be graphing a absolute value equation, graphing it, and showing it in piecewise notation.

y= |-3x+9|  <——- Note: absolute value brackets (|x|)

With a normal line, (-3x+9) it would go below the x-axis, but when there are absolute value lines in the equation it makes it so the line does not go into the negitive not. Which also means it does not go below the x-axis, istead it just rickashays off and comes right back up. It refects the original line and bounces off the x-axis.

 

Now that we haved graphed we must state it peace wise notation. The point that intercecpts with the x axis is where you will be basing the x is < or > off of. For stating the x is < or > is with two equations, becuase the graph shows two line reflecting off the x axis there are two equations. 1. -3x+9 and -(-3x+9) then you state whether  x is < or > for the number on the x axis.

y = {-3x+9  if x < and equal to 3}

y = {-(-3x+9)  if x > 3}

To figure out if x is < or > you must test a point on the x axis to see if its true. Test numbers before and after three.

Week 11/ Solving system of equations graphically

Solving system of equations

First you must be able to show the equation graphically. By doing so you have to show any linear equations into y=mx+b form; then change any quadratic equations into factored form or vertex form.  After you have algebraically changed all the equations you can then graph the equations and see where they intercept. Then that will be your solution.

There can be three type of solutions when solving for x equations that are linear and quadratic. The first one is no solution, the second is one solution, and the third is two solutions.

Equations: y= 2x^2 + 4x + 4

y= -2x+7

The graph states that there is no solution because the equation does not touch or intercept with each other.

Week 7 blog post/ Discriminant

Discriminant is used when figuring out how many solutions and if they are real roots. The Discriminant is b^2 – 4ac$, which looks familiar beucase it is the discriminant of the equation in the quadratic formula. Just by the simple equation b^2 – 4ac$ it can tell us alot about the quadratic equation before we even start solving.

If you get zero means x is one solution one solution, if the x is greater than zero you have two solutionsand you know it will be rational, if you get less than zero there is no solution.

Week 10 blog post/ Solving Quadratic Inequalities

In Solving Quadratic Inequalities, you need to know three steps.

Step 1: Factor the exspression

Step 2: Determine the zeros

Step 3: Use a sign chart for each factor to the left and right of zeros

Note: Test numbers a number on all the opposing sides of the x intercepts

(Key things to know when solving Quadratic Inequalities is to make sure the the statement is always true, if its not you know your answer is wrong or there is no possibility.)

Week9/solving A with factored form

Question: A graph passes through B(2,-5) and has x-intercepts -3 and 4

Before solving its good to analyse the question first. Passes through B(2,-5) represents x and y, well x intercepts represents the x intercepts. An equation that can use both of these clues is factored form since factored form has the x-intercepts in the equation.

Factored Form y= a(x-x1) (x-x2)

When plugging in the x intercepts from the equation you get  y= a(x+3) (x-4), this question may look solvable but it has more than one variables so you cannot solve it yet for A. Thats why you put the x and y into the equation. Now your left with -5= a(2+3) (2-4). From there you can now solve for a. (EXAMPLE BELOW)Now the equation is y= \frac{1}{2}(x+3) (x-4) now it is a complete equation and can be changed and graphed.

Week 8/Standard Form-Graphing

When your given an equation in General form, there isn’t enough information to easily graph just by looking at it. Your first step is to change the equation into Standard Form. To change a linear equation into Standard Form you can complete the square giving an answer presented into standerd form.

Note: Standared form is presented in a way to graph linear equations easily.

Example: y=x^2-4x+1

Standared form: y=(x-2)^2-3

After the equation has been changed into Standared Form the equation y=(x-2)^2-3 tells you everything you need to know to graph the equation. The equation is represented by y=a(x-b)^2-c, where A tells you if the parabola opens up or down and if it is congruent to x^2, B tells you where to move on the x axis, well c tells you where to move on the y axis.

Those are the basics to graphing and algebraically changing linear equations to standard form.

Week 6 / Quadratic Formula

The Quadratic Formula is a well known formuala used to solve quadratic equations; a quadratic equation is an equation (=) that has a squareroot symbole. To solve these you can use a number of things, but my favourite way is the Quadratic Formula.

To start you must know the Quadratic Formula which is x= -b +or-\sqrt{(b)^2 -4ac} then all of that divided by 2a. The A, B, and C, they all are pieces of the equation  x^2-6x+4=0  a=x^2 b=-6x c=4. The letters are just to show you where to place the numbers in the equation.

After you have plugged in the numbers the rest is pretty sympol, you just solve after that.

Week 5 / The Difference of Squares; Factoring

Difference of Squares / Factoring

To understand how to factor Difference of Squares you must first know what Difference of Squares means. Difference means in math “subraction.” Squares means two numbers will square into another number. Example ({3}^2 {3 x 3} makes 9)

Forthgoing, figuring out if something is a Difference of squares or not, here two examples.

Easy equations

First when factoring you must find what is common, I found that {5x}^2 is something i can pull out of the equation and then your left with {5x}^2 (3x-1). To figure out if its aDifference of squares you check if it is squared and has a negitive sign; so yes it is a difference of squares.

The second equation is much more different than the first equation, its a trinomial, but that doesnt make a huge differenence when it comes to factoring the equation. Since both equations are easy equations, just checking what is common is all we need to do. In the equation 4 is the only number that is common, and your left with 4({2p}^3{1p}^2 -1). Since that is all you can do now you must find out if its a difference of squares or not. You atuomatically know becuase the 2p is cubed. So it is not a difference of squares.

Week four blog post

Simple equations solving for X (with radicals)

1st step- Always the first step to solving with radicals is to find restrictions. Restrictions are a way to check that your answer is right by looking at the restrictions. When the equation that only has a radical and a coefficient infront, the same restriction applies. (x \geq 0)

2nd step- If there is a coefficient infront of the radical, do the opposite and divide the seven out from the radical. But watch out what you do to one side you must do it to the other; when you divided the seven, you have to divide the seven by the fourty two.

3rd step- Now that you are left with 6 = \sqrt{2x} to get rid of a square root symbole, your going to have to do the oppostite of square rooting, powering! You have to power the square root symbol, but what you have to do to one side, you have to do to the other. That means 6 turns into 36 becuase 6 times 6 is thrity six, well \sqrt{2x} becomes just 2x.

4rth step- Now that your just left with 36 = 2x, to seperate the 2 from the x you must to the oppostite and divide both the 2x and the 36 by the number attached to the x, in turn leaving the x by itself. So x = 18.

Finally- Your not done yet! Just becuase you have the asnwer doesnt mean your finished, you still have to check to see if you answer mathces your restrictions, if it doesnt its wrong. x \geq 0 does match x=18. That mean you got the right answer.