Which equation is y = (x + 3)2 + (x + 4)2 rewritten in vertex form?
Answers
Answer:
we have
y=(x+3)^{2}+(x+4)^{2}y=(x+3)
2
+(x+4)
2
\begin{gathered}y=(x+3)^{2}+(x+4)^{2} \\y= x^{2} +6x+9+x^{2}+8x+16 \\y=2x^{2}+14x+25\end{gathered}
y=(x+3)
2
+(x+4)
2
y=x
2
+6x+9+x
2
+8x+16
y=2x
2
+14x+25
rewrite now in vertex form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
y-25=2x^{2}+14xy−25=2x
2
+14x
Factor the leading coefficient
y-25=2(x^{2}+7x)y−25=2(x
2
+7x)
Complete the square. Remember to balance the equation by adding the same constants to each side
y-25+24.50=2(x^{2}+7x+3.5^{2})y−25+24.50=2(x
2
+7x+3.5
2
)
y-0.50=2(x^{2}+7x+12.25)y−0.50=2(x
2
+7x+12.25)
Rewrite as perfect squares
y-0.50=2(x+3.5)^{2}y−0.50=2(x+3.5)
2
y=2(x+3.5)^{2}+0.50y=2(x+3.5)
2
+0.50
the vertex is the point (-3.5,0.50)(−3.5,0.50)
therefore
the answer is
y=2(x+3.5)^{2}+0.50y=2(x+3.5)
2
+0.50
Answer:
This is a bit of a sneaky question. It isn't immediately obvious that this is a parabola, but "vertex form" is a form of equation specifically for one. It is a parabola, a closer look reveals, which is fortunate... It's the same thing as "completing the square" - we want the equation in the form
a
(
x
−
h
)
2
+
k
.
To get there from here, we first multiply out the two brackets, then collect up terms, then divide through to make the
x
2
coefficient 1:
1
2
y
=
x
2
+
7
x
+
25
2
Then we find a square bracket that gives us the correct
x
coefficient. Note that in general
(
x
+
n
)
2
=
x
2
+
2
n
+
n
2
So we choose
n
to be half our existing
x
coefficient, i.e.
7
2
. Then we need to subtract off the extra
n
2
=
49
4
that we've introduced. So
1
2
y
=
(
x
+
7
2
)
2
−
49
4
+
25
2
=
(
x
+
7
2
)
2
+
1
4
Multiply back through to get
y
:
y
=
2
(
x
+
7
2
)
2
+
1
2
Answer link