This is a blog for my second semester pre-calculus 30s students. Have a blast!

Tuesday, February 14, 2006

Valentine's Scribe

Hello!

It's true that we learn new stuff everyday. Today in Pre-Cal, we learned about determining characteristics of a quadratic function algebraically, or in other words, without you using a graphing calculator.
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DETERMINING CHARACTERISTICS OF A QUADRATIC FUNCTION ALGEBRAICALLY


First of all, we now know what is the standard form of a quadratic function. Now on this lesson, we used the expanded form mostly.

The expanded form of a quadratic function is:

y = ax2 + bx + c

1) a represents the width and direction of the opening of the parabola.
2)
b creates an oblique translation (a translation of a line that is neither parallel nor perpendicular)
3) c is basically the y-intercept of the parabola.

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Recall: Find the roots and the vertex of f(x) = 6x2 + x -12 using zero product property.

Factoring the expression:
f(x) =
6x2 + x -12
f(x) = (2x + 3)(3x - 4)

Ah.. this involves fractions. So this is how we set it up to find the roots:
2x + 3 = 0
| 3x - 4 = 0

(note: ALWAYS organize your solutions.)

Solving for the roots:
2x + 3 = 0 | 3x - 4 = 0
2x = -3 | 3x = 4
x = -3/2 | x = 4/3

So the roots of the equation f(x) = 6x2 + x -12 are:
roots:
x = -3/2, 4/3.

* the x coordinate of the vertex a.k.a horizontal shift is found by finding the midpoint of the roots:

h = (x1 + x2) / 2

To find the horizontal shift, we just need to substitute x values in the equation with the roots.

h = (-3/2 + 4/3) / 2

Ah... dissimilar fractions. Now we need to find the LCD of the equation so we can solve them. LCD = 6.

h = (-9/6 + 8/6)/ 2
h = (-1/6) / 2
h = -1/12

Now we have found the x coordinate a.k.a horizontal shift of the vertex. We just need to find the y coordinate a.k.a vertical shift so that we can find the vertex. To find the horizontal shift, we substitute
"h" back into f(x).
[*Note: This involves ugly fractions for this particular solution, so please just bear with me.]

And oh, don't make it hard for yourself, just use the factored form of the expanded form of the parabola since it's easier. This is one handy tip coming from me. ;)

f(-1/12) = [2(-1/12) + 3][3(-1/12) + 4]
f(-1/12) = [-1/6 + 3][-1/4 + 4]
f(-1/12) = [-1/6 + 18/6][-1/4 + 16/4]
f(-1/12) = [17/6][17/4]
f(-1/12) = -289/29

Now we have the y coordinate of the vertex a.k.a horizontal shift. Therefore, we can state that:
vertex: (
-1/12, -289/29)

ex. Find the roots, vertex, axis of symmetry of f(x) =
x2 + 4x -5

Solving for the roots:
x2 + 4x -5
0 = (x + 5)(x - 1)

x + 5 = 0 | x - 1 = 0
x = -5 | x = 1

Therefore, we can say that the roots:
-5, 1.

Finding the x coordinate of the vertex a.k.a horizontal shift:

h = (
x1 + x2) / 2
h = (-5 + 1) / 2
h = (-4) / 2
h = -2

Now that we know the x coordinate of the vertex a.k.a horizontal shift, we can also state the axis of symmetry which is:
x = -2

Finding the y coordinate of the vertex a.k.a vertical shift:

f(-2) = (-2 + 5)(-2 -1)
f(-2) = (3)(-3)
f(-2) = -9

And there you have it! We can now state the coordinates of the vertex:
v:(
-2, -9)

We also came up with the rules of finding the roots, axis of symmetry, and vertex of a quadratic function in expanded form.

1) To find
roots:
- factor
y = ax2 + bx + c using zero product property

2) To find axis of symmetry, find the midpoint of the two roots. MAKE SURE to always write it like this:
x = h

3) To find vertex, substitute h for f(x) to get the k value. MAKE SURE to enclose the points of the vertex in round brackets as it is a point in the line.
vertex:(
h, k)
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And there you have it! That's today's lesson. Sorry it took a long time to post. Homework for today is Exercise 4, and tomorrow's scribe is..... (drumroll please)......
tim-Math-y. Haha.

Have a great evening people! Get out there and commit heinous acts of kindness.. Cheers!

6 Comments:

Blogger tim-Math-y said...

OMG! JOHN! HOW COULD YOU! lol.. grr.

8:14 PM

 
Blogger John D. - #12 said...

1) a represents the width and direction of the opening of the parabola.
2) b creates an oblique translation (a translation of a line that is neither parallel nor perpendicular)
3) c is basically the y-intercept of the parabola.

note: I found this in last semester's Pre-Cal 30S. It was posted by CraigK, so I gave him some credit for that. Mr. M didn't discuss much about this last class, so I guess I'd just post it for added information.

8:19 PM

 
Blogger vincent said...

bravo!

9:23 PM

 
Blogger Mr. Malandrakis said...

John, that was fantastic. Although so far this is what I've come to expect of you, so keep raising the bar. Great job!!!!

8:22 AM

 
Blogger Jojo said...

How can we top that John wow yo!

9:04 AM

 
Blogger Richy said...

Hahahaha. Excellent Scibe Post. Love the ending. Hahaha.

11:53 AM

 

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