The tutor gives another example of converting a quadratic function from standard to vertex form.  Some people find the case a<0 to be tricky.

 
My previous couple of posts (here and here) cover other examples of standard form to vertex form. Today’s focus:

Example: Convert y=-5x² + 30x + 11 to vertex form.

Solution:

Step 1: As before, rewrite the equation with a space before the constant term.

y=-5x² + 30x      +11

Step 2: Factor the coefficient of x² from both variable terms:

y=-5(x² -6x     ) +11

Step 3: As before, we complete the square (see here for details): we take half the coefficient of the x term, square it, then add that result inside the brackets.

In this case, the coefficient of x is -6. Half is -3, then (-3)² is 9.

y=-5(x² -6x +9) +11

Step 4: Very important: Adding the number inside the brackets actually means adding that number times the number in front. To equalize, we must add the opposite to the outside.

In this case, adding 9 inside the bracket, I’ve really added -5(9)=-45 to the equation. To equalize, I add 45 to the outside.

y=-5(x² -6x +9) +11 +45

Step 5: Realize that the three terms inside the brackets constitute a perfect square trinomial. (Once again, see this earlier post for more detail.)

In this case,

y=-5(x² -6x +9) +11 +45 becomes y=-5(x-3)² +56

For a reader who might still be unsure about this process, here’s proof that it’s valid:

y=-5(x-3)²+56

(x-3)² = (x-3)(x-3) = x² -3x -3x +9 = x² -6x +9 (by the foil method)

y=-5(x² -6x +9) + 56

Distribute the -5 into the brackets:

y=-5x² +30x -45 + 56

Now simplify:

y=-5x² +30x + 11

We arrive at the original equation, which means that our conversion of is valid.

Therefore, y=-5x² +30x + 11 becomes y=-5(x-3)² + 56 in vertex form. Its vertex is (3,56). (See my post here about identifying the vertex.)

Converting y=ax² + bx + c to y=a(x-p)² + q can be counter-intuitive when a<0. In particular, step 4 above may be difficult to believe at first. In a future post I'll likely visit another case of converting from standard form to vertex form. However, among this post and the previous two, the most important cases have been covered. HTH:) Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.