SME
Grade 9 Math · Problem type
Standard form: the vertex from −b/(2a)
Use this for “Find the vertex and the zeros of y = x² − 6x + 5.”
Chart
Vertex (3, −4), axis of symmetry x = 3
Zeros: x = 1, x = 5
y-intercept (0, 5)
f(6) = 5
This is an example. Type your own values to start a new problem.
y
=
a
x²
+
(
b
)
x
+
c
h
Vertex x
worked out (not typed)
k
Vertex y
worked out (not typed)
x₁
Left zero
worked out (not typed)
x₂
Right zero
worked out (not typed)
x
Input
example
y
Output
calculated
Formulas
h
=
−
b
2 ×
a
3
=
−
−6
2 × 1
k
=
a
×
h
2
+
b
×
h
+
c
−4
=
1
×
3
2
+
(−6)
×
3
+
5
x₁
=
h
−
√(−
k
a
)
1
=
3
−
√(−
−4
1
)
x₂
=
h
+
√(−
k
a
)
5
=
3
+
√(−
−4
1
)
y
=
a
×
x
2
+
b
×
x
+
c
5
=
1
×
6
2
+
(−6)
×
6
+
5
Assumptions
•
In y = ax² + bx + c the axis of symmetry is x = −b/(2a).
•
Put that x back into the rule to get the vertex’s y.
•
c is the y-intercept, the value at x = 0.
Step-by-step
WE KNOW
x = 6, a = 1, b = −6, c = 5
FIND
vertex x (h), vertex y (k), left zero (x₁), right zero (x₂), output (y)
Step 1 · Find vertex x (h)
Use h = −b ÷ (2 × a)
The
axis
of
symmetry
is
x
=
−
b
/(
2
a
),
halfway
between
the
zeros.
h
=
−
−6
2 × 1
h
=
6
2
h
=
3
Step 2 · Find vertex y (k)
Use k = a × h² + b × h + c
The
vertex
is
on
the
graph:
put
x
=
h
into
the
rule.
k
=
1
×
3
2
−
6
×
3
+
5
k
=
9
−
6
×
3
+
5
k
=
9
−
18
+
5
k
=
−4
Step 3 · Find left zero (x₁)
Use x₁ = h − √(−k ÷ a)
Set
y
=
0:
a
(
x
−
h
)
2
=
−
k
,
so
x
−
h
=
±√(−
k
a
).
The
minus
gives
the
left
zero.
x₁
=
3
−
√(−
−4
1
)
x₁
=
3
−
√
4
x₁
=
3
−
2
x₁
=
1
Step 4 · Find right zero (x₂)
Use x₂ = h + √(−k ÷ a)
The plus gives the right zero, as far right of the axis as the left one is left of it.
x₂
=
3
+
√(−
−4
1
)
x₂
=
3
+
√
4
x₂
=
3
+
2
x₂
=
5
Step 5 · Find output (y)
Use y = a × x² + b × x + c
Put the input into the rule: square it first, then multiply and add.
y
=
1
×
6
2
−
6
×
6
+
5
y
=
36
−
6
×
6
+
5
y
=
36
−
36
+
5
y
=
5
Check
Put the numbers back in. Both sides should match.
3
=
−
−6
2 × 1
✓
−4
=
1
×
3
2
−
6
×
3
+
5
✓
1
=
3
−
√(−
−4
1
)
✓
5
=
3
+
√(−
−4
1
)
✓
5
=
1
×
6
2
−
6
×
6
+
5
✓
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