Question

What is the equation of the axis of symmetry of the graph of y + 3x – 6 = –3(x – 2)squared + 4?

Answer 1

Answer:

x = 3/2

Step-by-step explanation:

That is quite long and drawn out, so we have to get it down to a single quadratic equation, set equal to 0.

Begin by expanding the right side to get

$y+3x-6=-3(x^2-4x+4)+4$

then multiplying in the -3 to get

$y+3x-6=-3x^2+12x-12+4$

Now we will combine all the like terms and get everything on one side of the equals sign, and set it equal to 0:

$-3x^2+9x-2=0$

In order to find the equation of the axis of symmetry we have to put it into vertex form, which is accomplished by completing the square on this quadratic.

In order to complete the square, the leading coefficient has to be a +1.  Ours is a -3, so we will factor that out.  First, though, now that it is set to equal 0, we will move the constant, -2, over to the other side, isolating the x terms.

$-3x^2+9x=2$

Now we can factor out the -3:

$-3(x^2-3x)=2$

The rules for completing the square are as follows:

Take half the linear term, square it, and add it to both sides.  Our linear term is 3.  Half of 3 is 3/2, and squaring that gives you 9/4.  We add into the parenthesis on the left a 9/4, but don't forget about that -3 hanging around out front that refuses to be ignored.  We didn't add in just a 9/4, we added in (-3)(9/4) = -27/4:

$-3(x^2-3x+\frac{9}{4})=2-\frac{27}{4}$

In the process of completing the square we created a perfect square binomial on the left.  Stating that binomial and simplifying the addition on the right gives us:

$-3(x-\frac{3}{2})^2=-\frac{19}{4}$

We can determine the axis of symmetry at this point.  Because this is a positive x-squared polynomial, the axis of symmetry is in the form of (x = ) and what it equals is the h coordinate of the vertex.  Our h coordinate is 3/2; therefore, the axis of symmetry has the equation x = 3/2