This week in PreCalc 11 we learned the properties of quadratic functions and how to analyze the quadratic equation. We learned that when are given the quadratic equation in vertex form then we can graph the equation.

Last unit we learned the quadratic equation ax^2 + bx + c = 0, this unit we learned y = a(x-p)^2 + q. This is combination of the equations y = (x-p)^2, y = x^2, and y = x^2 + q. These equations are all very important in graphing the curve the equations make.

Example 1:

y = x^2 (parent function: creates the most basic parabola)

y = x^2 + 3 (addition to the parent function)

Explanation 1:

What this equation tells us is based on the +3 at the end of it. This +3 is the vertical translation which means its moving the vertex from (0,0), as it would be in the equation y = x^2 because the +3 is replaced with a O, to (0, 3). It moves the y-intercept, in other words, it moves the vertex up or down. If the equation was y = x^2 - 3 then it would move the vertex down to (0, -3).

*vertex is either the highest point or lowest point of the parabola

*parabola is the curve the equation makes

*y-intercept is the point where the parabola crosses the y axis

Example 2:

y = (x - p)^2 y = (x-3)^2

Explanation 2: 

The part that tells us about the graph is the -3. This -3 is the horizontal translation which means it moves the vertex from (0,0) to (3,0). It moves the x-intercept, in other words, it moves the vertex left or right. the reason that it is a positive three, but the equation has a negative 3 is because when you place a +3 in the equation y = (x-(+3)^2, the negative overtakes the positive. If it was a -3 then two negatives make a positive so it would be y = (x+3)^2

*x-intercept is where the parabola crosses the x axis

Example 3:

y = ax^2

y = 2x^2 (tall and skinny)

y = 1/2x^2 (wide)

Explanation 3:

The 2, in this case, is telling us about the stretch reflection which means that it tells us how wide the parabola will be or how skinny it will be. If the a value is greater than 1 then it’s tall and skinny. If a is less than one and greater than 0 then it’s wider.

That’s why when we are given an equation in vertex form ( y = a(x-p)^2 +q ) then we can easily graph it. It’s called vertex form of the equation because you are given the vertex (the opposite sign of the p value is the x value and the q is the y-value). From the equation we are also given the stretch value which tells us how wide or skinny the parabola is going to be and also tells us the pattern we need to follow to get the correct function. The parent function,y = ax^2 gives us the vertex (O,O), and it tells us to follow the pattern up 1, over 1, up 3 over 1, up 5 over one, and it keeps going up by 2. But when there is a stretch value that isn’t one then that pattern changes. If the a value is 2, for example, then you multiply the basic pattern by 2 so the pattern is up 2 over 1, up 6 over 1, up 10 over one, and so on. If the a value is, for example, 1/2 then you multiply the basic pattern by one half so it would be so that you go up 1/2 over 1 and so on.

Example:

So if the equation is: y = 3(x-6)^2 +4

That’s how a quadratic equation in the vertex form is graphed.