# Week 8 – precalculus 11

Transforming the Graph of $y=x^2$

There are three equations:

$y=x^2 + q$

$y=(x-p)^2$

$y=ax^2$

The first one adds q to the parent function of $y=x^2$, this decides the Y intercept and so the parent either goes up or down when this is applied. The second one changes the x intercept and there changes the parabola side to side, if p is being taken away the parabola goes to the right and the opposite happens when it is added. The third one adds an a to the parent function. This decides the strech factor, if the stretch is a higher number the parabola is skinnier. If a is negative the same is true for more negative numbers, the parabola also opens downward if a is negative.

I will show four examples to explain this, using desmos. I will compare it to the parent function, which will be red. All of these formulas can be added together to form $y= a(x-p)^2 + q$.

$y=x^2 + 4$

As you can see, the y intercept is changed to +4 and therefor the parabola moves up 4, no other changes are made.

$y=(x-3)^2$

The x intercept is being changed and the parabola moves 3 to the right because we are taking away p

$y=2x^2$ and $y=\frac{1}{2}x^2$, in this example I will show two variations. The fraction is blue and the 2 is green.

As you can see, the fraction is wider than the parent, but the 2 is skinnier. The stretch value only changes the width of the parabola. The parent values go up by 1,3,5,7 and over one when creating the line. For the 2, the values go up by 2,6,10,14 over 1 each time. For the fraction the line goes up by 1/2, 3/2, 5/2, 7/2, causing the line to be wider because it goes up by less but still goes over 1 each time.

$y= -2(x+4)^2 + 3$.

The parabola moves up 3 because the y-int/q is changed and the p values has caused the line to move 4 to the left. The a value is negative so the parabola opens downwards, it is also lower that -1 sow the parabola is skinnier.

Each of these variables changes the parabola in its own way but we can combine all the equations together to make many changes at once. You can even find the vertex by using the standard formula. By knowing the p and q values, because the line parabola will always be how much higher/lower it is on the y axis and how far left or right on the x axis.