Answer:
To graph a quadratic equation we first need to factorise it into a different form.
First we check what the discriminant is equal to
Where
f
(
x
)
=
a
x
2
+
b
x
2
+
c
Δ
(Discriminant)
=
b
2
−
4
a
c
In this case
Δ
=
3
2
−
4
⋅
2
⋅
7
Δ
=
−
47
Because it is less than zero it can't be factored normally
Therefore we must use the The Quadratic Formula or Completing the Square
Here I have completed the square
f
(
x
)
=
2
x
2
−
3
x
+
7
Remove factor from
x
2
term
f
(
x
)
=
2
⋅
(
x
2
−
3
2
x
+
7
2
)
Take
x
term, half it and then square it
f
(
x
)
=
−
3
2
→
−
3
4
→
9
16
Add and then subtract this number inside the equation
f
(
x
)
=
2
⋅
(
x
2
−
3
2
x
+
9
16
−
9
16
+
7
2
)
Combine the first three terms in a perfect square
f
(
x
)
=
2
⋅
(
(
x
−
3
4
)
2
−
9
16
+
7
2
)
Equate left over terms
f
(
x
)
=
2
⋅
(
(
x
−
3
4
)
2
+
47
16
)
Multiply coefficient back in
f
(
x
)
=
2
(
x
−
3
4
)
2
+
47
8
This gives a turning point of
(
3
4
,
47
8
)
=
(
0.75
,
5.875
)
and a
y
intercept of
2
⋅
(
3
4
)
2
+
47
8
=
(
0
,
7
)
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