Changing from standard form to vertex form-Quadratic Equations
Standard form = ax^2 +bx+c=y
Vertex form = a(x-h)^2 +k =y
The video gives step by step directions for converting from standard to vertex form.
Hi welcome to MooMooMath Today we are going to talk about Quadratics and changing a quadratic from standard form to vertex form Let's overview quickly what standard form is. Standard form is ax squared plus bx plus c is equal to y ( ax^2 +bx +c = y) and vertex is equal to a parenthesis x minus h quantity squared plus k equals y. ( a(x-h)^2 +k=y ) Those are the two forms of the quadratic. The vertex form is handy because the h k is your vertex and standard form is nice if you are using the quadratic formula. Sometimes we have to go from standard form to vertex form . Let's learn how to do that. Let's start with a lead coefficient of one which is the easier one. We have x^2 + 8x + 3 = y We want to change this to the vertex form. Step 1 is to group our x values together and complete the square.so we can write the equation as a perfect square. I will write the x^2 and 8x together. I will push the positive 3 to the side because the constant 3 does not complete the square. What we need to do is find a value that completes the square. I will take the value b which is the coefficient to the linear term and half it and square it and this will complete the square. I will take b and half it and then square it. Half of 8 is 4 and 4 squared is 16. I will put 16 back in the equation to complete the square. What I have completed with these three terms is a trinomial that will factor to (x+4)^2 Next I can't just add 16 to the equation because it will be unbalanced. When I add 16 it is out of balance, so I have to subtract 16 from the equation to get it back in balance. I will group negative 16 with constant 3 and add these together to get -13. What I have done is combine those two constants together, and now we have our quadratic in vertex form. h is -4 and k is -13. h is always the opposite sign of what we see in the equation because it is x-h and k is the same value which is -16. The vertex is (-4,-13)
Let's try another one so you can see the pattern.
x^2 + 24x -1 = f(x)
Group the x's together x squared and 24x
push the 1 over which will become part of my k value. Part of the constant on the outside.
Now complete the square, take 24 and half it and square it So half of 24 is 12 and 12 squared is 144. We can't just 144 without subtracting 144 . Now group the first three terms together to make our perfect square. The perfect square it factors to is always the square root of what we just found. (the 144) and in the back we get negative 145 . Now we have our functions, so we can figure out our h and k. h is the opposite of what we see (-12) and k is the same sign as what we see, and there is your vertex that you can use to graph your quadratic equation. Hope this video was helpful.