In this unit we learned how to graph a parabola using a quadratic function. There are three different forms that you can write a quadratic function: Vertex form which is y=a(x-p)²-q, Standard form which is y=ax²+bx+c, the last one is Factored form which is y=a(x1-x1)(x2-x2).
All three functions gives you different information. The vertex form gives you the most important part of the parabola which is the vertex, using the vertex you can find the line of symmetry so you can say that it gives you the line of symmetry, and the stretch. The standard form gives you the y intercept and the stretch. The factored form gives you the x intercepts and the stretch.
Sometimes you will be required to convert an equation to a different form in order to find a different piece of information. So In order to go from standard form to vertex form you need to complete the square. In order to go from vertex form to standard form you need to expand. In order to go from standard form to factored you have to factor. In order to go from factored form to standard you need to expand.
vertex form <—-> Standard form <—-> Factored form
You cannot go from factored form to vertex form and vice versa
Vertex Form: y = 8 (x – 7)² – 4
Standard Form: y = 8x² – 112x +388
Factored Form: Not factorable
Line of symmetry: x = 7
Y intercept: (0,388)
X intercept: (6.293,0) , (7.707,0)
In this unit we learned how to factor trinomials, complete the square, and the quadratic formula in order to find the x intercepts. With completing the square and the quadratic formula there was many places where you could make silly mistakes so it was best to go slow while really thinking.
I will be explaining completing the square, I struggled with completing the square the most out of the three different ways to solve for x. There can be fractions which was where I made the most mistakes. So the steps to complete the square are fairly simple. with an equation like x²+2x+4=0 you need to first divide the second term by 2 after dividing the second term by 2 you need to square the quotient and make a 0 pair.
so in this case: (x²+2x+1-1)+4=0
Then you have to start isolating x: (x+1)²+3=0 —> (x+1)² = -3
Then you square root both sides to get rid of the exponent, which makes the equation: x+1= +-√-3
In this case the number under the square root is a negative so it is impossible to do because there are no real roots. I stopped there because I noticed the negative but if it were a positive you would isolate x by subtracting the 1 and moving it to the other side. Then you would proceed to solve the equation.
Now if there was a leading coefficient greater than 1 than you have to divide everything by the leading coefficient to get x by it self.
ex: 2x²+4x+10=0 —–> x²+2x+5=0
Then you do the same process as I showed earlier
For the second unit in precalculus 11, we learned how to multiply and divide radicals. In order to solve these equations we had too factor and we could verify our solutions. I will be demonstrating how to make an entire radical into a mixed radical, this was used in order to make the equation a lot easier. In order to make an entire radical into a mixed you need to find the factors (if it is square rooted you need to find the perfect square factors, and if it is cube rooted, you need to find the perfect cube factors.) here is an example:
Now that I have explained how to convert a entire radical into a mixed radical I will show explain how to convert a mixed radical into an entire radical. So when you have a mixed radical and you want to have the entire radical you first need to square the coefficient and then once you have done that then all you need to do is multiply the two square roots as you would with regular numbers and you will have your entire radical. example:
This is my unit summary assignment for precalculus 11 (roots and powers)