Download Converting Between Forms of a Quadratic Equation

Converting Between
Forms of a
Quadratic Equation
I. Converting Vertex to Standard form
• To convert from vertex to standard form you
just follow the order of operations to remove
the ( ) from the equation.
• Remember PEMDAS…. Exponents come
before multiplication!!
Let’s work through an example…
Convert y  2( x  3)2  1 to standard form.
Step 1: Rewrite the equation as follows…
y  2 ( x  3)( x  3)  1
Because to square something means to multiply
it by itself
Now, FOIL the binomials…
y  2  x  3x  3x  9   1
2
Then combine like terms inside the brackets…
y  2  x  6 x  9   1
2
Next step… Distribute the 2 inside the
brackets… NOT to the +1
y  2  x  6 x  9   1
2
That gives us…..
y  2 x  12 x  18  1
2
Finally, combine like terms
y  2 x  12 x  19
2
This is in standard form!
Every vertex to standard form
problem will follow the same
pattern of steps…
Try this one….
• Convert to standard form…
y  ( x  4)  6
2
Answer:
y    ( x  4)( x  4)   6
Rewrite
y    x 2  4 x  4 x  16  6
FOIL
y  [ x 2  8 x  16]  6
Combine like terms
y   x 2  8 x  16  6
Distribute
y   x 2  8 x  10
Combine like terms
II. Standard to vertex form
• The goal is to rewrite
y  ax  bx  c as
2
y  a( x  h)  k
2
• To be able to write the vertex form you need
to know the…. Vertex!
So find the vertex of the parabola the way you
do to graph the equation: x=-b/2a then plug that
value in to get the y-coordinate.
Let’s do one together…
2
y

2
x
 4 x  7 to vertex form.
Convert
STEP 1: Find the vertex.
x
b ( 4) 4

 1
2a
2(2) 4
Now plug x=1 into the equation to find y.
y  2(1)2  4(1)  7  2  4  7  5
So the vertex is (1,5).
In vertex form the vertex is (h,k)…
so if we know the vertex is (1,5)
that means that h=1 and k=5. So
let’s substitute those values into
our vertex form…
y  a( x  h)  k
2
Becomes…
y  a( x  1)  5
2
Now the only thing left to find is the “a”
value. Good news, this is easy! What is
“a” in the standard form? Remember,
2
“a” is in front of the x term.
y  2 x2  4 x  7
So a=2… The “a”’s are the same in both
forms so the “a” for our vertex form is
also 2. Then the vertex form becomes
y  2( x  1)  5
2
And we’re finished!
Try this one…
Convert to vertex form.
y  3x  6 x  4
2
Find the vertex.
b (6) 6
x


 1
2a 2(3)
6
y  3( 1)2  6( 1)  4  7
So the vertex is (-1,-7) which means
h=-1 and k=-7.
The “a” value in standard form is 3
so the “a” in vertex form is 3 also.
That means our vertex form is ….
y  3( x  ( 1))  7
2
Now, rewrite with out the double
sign…
y  3( x  1)  7
2