SME
Grade 9 Math · Problem type
Factored form: the zeros
Use this for “Graph y = −2(x − 1)(x − 5): its zeros, axis, vertex and y-intercept.”
Chart
Vertex (3, 8), axis of symmetry x = 3
Zeros: x = 1, x = 5
y-intercept (0, −10)
This is an example. Type your own values to start a new problem.
y
=
a
(x
−
p
)(x
−
q
)
h
Axis of symmetry x
worked out (not typed)
k
Vertex y
worked out (not typed)
c
y-intercept
worked out (not typed)
Formulas
h
=
p
+
q
2
3
=
1 + 5
2
k
=
a
×
(
h
−
p
)
×
(
h
−
q
)
8 = −2 × (3 − 1) × (3 − 5)
c
=
a
×
(0
−
p
)
×
(0
−
q
)
−10 = −2 × (0 − 1) × (0 − 5)
Assumptions
•
y = a(x − p)(x − q) is 0 when x = p or x = q: a product is 0 only when a factor is 0.
•
The axis of symmetry is halfway between the zeros.
•
Put the axis’s x into the rule to get the vertex’s y.
Step-by-step
WE KNOW
a = −2, p = 1, q = 5
FIND
axis of symmetry x (h), vertex y (k), y-intercept (c)
Step 1 · Find axis of symmetry x (h)
Use h = (p + q) ÷ 2
The axis is halfway between the zeros: their mean.
h
=
1 + 5
2
h
=
6
2
h
=
3
Step 2 · Find vertex y (k)
Use k = a × (h − p) × (h − q)
The
vertex
is
on
the
axis:
put
x
=
h
into
the
rule.
k
=
−2
×
(3
−
1)
×
(3
−
5)
k
=
−2
×
2
×
(−2)
k
=
−4
×
(−2)
k
=
8
Step 3 · Find y-intercept (c)
Use c = a × (0 − p) × (0 − q)
Put x = 0 into the rule for the y-intercept.
c
=
−2
×
(0
−
1)
×
(0
−
5)
c
=
−2
×
(−1)
×
(−5)
c
=
2
×
(−5)
c
=
−10
Check
Put the numbers back in. Both sides should match.
3
=
1 + 5
2
✓
8 = −2 × (3 − 1) × (3 − 5) ✓
−10 = −2 × (0 − 1) × (0 − 5) ✓
Topic check
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