Week 9 in Precalc 11

This week in math we learned how to convert quadratic functions into general form, factored form, and standard form from other forms. General form is y=ax^2+bx+c, where c is the y-intercept. Factored form is y=(x-x1)(x-x2), where x1 and x2 are the x-intercepts. Standard form is y=a(x-p)^2+q, where a is the stretch or compression of the parabola and whether its a minimum (opens up) or maximum (opens down) vertex. P is the horizontal translation, moves vertex left (+) or right (-) on the x-axis. Q is the vertical translation, moves vertex up (+) or down (-) on the y-axis.

To start with the easiest conversions first, to go from factored form to general form, you simply expand out. The same goes for standard form to general form, you simply expand out. To go from general form to factored form, you factor it. You can do this by using the CDPEU (common? Difference of squares? Pattern? Easy? Ugly) method we previously learned about, remember for ugly trinomials, the box method is easy to use. If it simply doesn’t factor, then use the quadratic formula.

You can’t convert factored form directly to standard form, so going onto our last conversion. To convert from general form to standard form, you will complete the square. We’ve previously done this when trying to find the x-intercepts, but this time you will only go so far as to when it’s in standard form, so y=a(x-p)^2+q.

The first step, as a refresher, is to remove the coefficient on the first term from the first two terms if there is one, if not ignore this step. Don’t remove the coefficient from the last term, leave it outside of the brackets. Next, you add +__-__ after the second term in the brackets (or not if there is no coefficient on the first term). Make sure the positive is first! You will find the number that goes into these spaces by taking the second term in the brackets and dividing it by two then squaring it, (a/2)^2, a = second term number. The sign in front of the number doesn’t matter.

Now, you will factor the trinomial in the brackets (first three terms) and multiply the coefficient by the last term in the brackets so it can leave the brackets. Then you combine the two numbers outside the brackets (add or subtract) and then it is standard form, y=a(x-p)^2+q.

Here are some examples below of how to convert from form to form:

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