Week 8 – Pre-Calc 11

Oh boy, coming back from spring break was not easy. However, I still managed to learn a whole lot from this week. We are now graphing up a storm with quadratics. This is the most times in my life I’ve used the word parabola. Speaking of “parabola” that’s this weeks topic!

First of all, a quadratic function includes a which will graph out a parabola. A parabola is a big U shape that constantly slopes outward like this: 

The parent function, or where all parabolas start is From this form, we can add variables to say whether the parabola will be moved up or down, side to side, which way it faces and how skinny or thick it will be.

For the up or down (y-axis) there will be a plus or minus added to the end of the equation such as This means the parabola will be 5 up on the y-axis but still at zero on the x-axis.

Now to change the side to side (x-axis), we will add brackets in like this The MOST IMPORTANT thing to remember here is that this is NOT 7 to the left. A negative goes to the right and a positive goes to the left. This is a rule that only applies in the brackets as you can see compared to the example 1.

To show which way the parabola opens, there will either be a nothing or a negative in front of the x value like this  In this case with a negative, the parabola will open downwards.

The thickness and skinniness come from a coefficient in front of the x value. Here is an example  very skinny compared to 

Now you bring all these factors together and you have standard form which is 

a= Stretch/Compression (Thickness/Skinniness), p= horizontal translation (side to side) and q= Vertical translation (Up and down)

A bonus is that when you put your two translations together, you get a vertex! A vertex is an absolute minimum or maximum point on a parabola. Take 

There is no coefficient, so the parabola will be a normal parabola facing upwards and it will be moved 1 unit to the right and 2 units up. These x and y values give you a vertex of (1,2). Even better, you can also find your line of symmetry which is the line that goes up the middle of the parabola. In this case, it’s 1.

Of the 7 key characteristics of a parabola, we’ve covered, vertex, maximum/minimum and line of symmetry.

(7 key characteristics): Vertex (the most important), x-intercept(*s), y-intercept, Domain, Range, Maximum/Minimum and the line of symmetry.

All that’s left are the x and y-intercepts which are where the lines of the parabola cross the respective axises.

We also have the domain and range which is just between which numbers will the variables fall. So far x is almost always real and y will be bigger or smaller than a given value.

That’s all for this week, I hope you enjoyed.