Section+1.5

Section 1.5 focuses on Quadratic Functions in various forms, their graphs and transformations from the graph of f(x) = x².

Terminology: Quadratic Function:** has an x and y variable with the highest power of x being 2 ex. f(x) = x² + 3x + 7 yo! f(x) = ax² + bx + c examples: f(x) = 4x² + 3x - 5 where a, b and c are constants h(x) = -x² - 7x + 3 a cannot equal zero
 * [[image:arch.jpg width="312" height="227"]][[image:archway.jpg width="321" height="229" align="right"]]
 * Parabola:** the U shaped graph of a quadratic function.
 * Quadratic Equation:** an equation with just x variables with the highest power of x being 2 ex. 0 = x² - 3x + 4
 * General Form of a Quadratic Function **

f(x) = a(x - h)² + k examples: f(x) = 3(x - 4)² + 5 where a = 3; h = 4; k = 5 where a, h and k are constants h(x) = -2(x + 3)² -4 where a = -2; h = -3; k = -4 a cannot equal zero
 * Standard Form of a Quadratic Function **

The three constants in Standard Form can be used to TRANSFORM the graph of y=x² ** a→ ** represents the vertical strecth(narrower) or shrink(wider) if a › 1; then the graph is stretched if 0‹a‹1; then the graph is shrunk if a is negative, the graph is flipped upside down (opens down)
 * h→ **represents the horizonal shift of the vertex of the graph (right if positive, left if negative)
 * k→ **represents the veritcal shift of the vertex of the graph (up if positive, down if negative)

We should have the graph of f(x) = x² memorized.


 * Vertex: ** is the lowest point on the parabola if the graph opens up or the highest point on the parabola if the graph opens down. In general form, the vertex is represented as (-b/2a, c - b^2/4a). In standard form, the vertex is expressed as (h, k). Therefore, converting a quadratic function from general to standard form by completing the square may save time and work.


 * Line of Symmetry: the equation for the line of symmetry is -b/2a.**

The Line of Symmetry can also be expressed as the x value in the ordered pair. For Example, in the ordered pair (3,-2) the axis of symmetry would be -2 because if you could imagine a vertical line running through the x point than it would equal the axis of symmetry. media type="custom" key="4514538"

The method of changing a quadratic function from General form to Standard form includes compeleting the square.
 * How to change a quadratic function from Standard to General Form? **

SEE ATTACHED EXAMPLE...

Completing The Square
First, let's take a sample equation. We'll try to put into standard form

There are four steps to completing the square:


 * 1)** First, the leading coefficient (**a**) must be 1. In some problems it may already be 1, but in the problem above, we'll have to factor it out (of the first two terms). After we factor out the 2, it should look like: [[image:equation(7).png]]


 * 2)** Second, the **b** value will be divided by two. In this case, our **b** value is 3. Half of 3 is 1.5, but in a problem like this, it's much easier to write in fraction form: [[image:equation(8).png]] (keep in mind that the number you got from the division will be the **h** value in the final equation).


 * 3)** Third, you square the number you got from the division and add it to the equation. Remember that when you square a fraction, you take the square of the numerator and the square of the denominator: [[image:equation(10).png]]. Our equation should now look like [[image:equation(11).png]].


 * 4)** Because we added a number to the equation, we have to subtract that number from the same side to keep the total values on either side of the equal sign balanced. Remember that we didn't just add 9/4, we added 9/4 times two. Our equation should now look like this: [[image:equation(12).png]] .Now, factor the equation. It should now look like: [[image:equation(15).png]]

With the last two numbers worked out, our final result should come to:


 * Finding Zeroes in a Quadratic equation**

1. The zeroes of a function are those point on a parabola where y = 0. (or where the parabola crosses the x axis) 2. To find these points, the equation must be in general form 3. The simples way to find the zeroes is to factor this formula. if the formula is factorable, take the oppisite of the numbers in the parenthesis Ex: When this is factored, it is show as, In this equation, there is one zero at -2.

4. If an equation does not factor, you can use the quadratic formula: In doing so you will find 2 solutions because you are either adding or subtracting the radical. These 2 solutions are your zeroes.


 * Completing the Square Example!!!!!!!!!!!!**

y = x^2 + 10x + 1 y - 1 = x^2 + 10x y - 1 = x^2 + 10x + 25 y +24 = (x + 5)^2 y = (x + 5)^2 - 24

Vertex= (-5,-24) Axis of Symmetry = X = -5

By: **MIIICKKK ;)**


 * Word Problem Example**

a rectangular parking lot, that is closed off by a wall on one side, has a 500ft fence around it. What is the measurement of the widths and length of the rectangles sides and what is the max area inside the fence.

(500-2x)(x)= -2x²+500x = 2x²-500x = 2(x²-250x) = 2(x-125)²+31250 A=2 H=125 K=31250

Width= 125 ft Length= 250 ft Area= 31250 ft²

(Alex Pizzala) There is a piece of pipe that is cut into two pieces. The total sum of both pieces squared is 202. Find the two pieces. -We could call one piece (x). The other would be (20-x) x² + (20-x)² = 202 x² + (20-x) (20-x) - 202 = 0 x² +400 - 40x + x² - 202 = 0 2x² -40x + 198 = 0 x² - 20x + 99= 0 (x - 11) (x - 9) = 0 x=11 ; x=9