SME
Grade 9 Math · Problem type
Transform a function: g(x) = a·f(x − h) + k
Use this for “f(x) = x². Graph g(x) = 2f(x − 3) + 1. Where does the point (1, 1) of f go?”
Chart
(1, 1) on f moves to (4, 3) on g: x + 3, then 2 × y + 1
This is an example. Type your own values to start a new problem.
g(x)
=
a
f(x
−
h
)
+
k
p
Point of f: x
example
q
f(p) = p²
worked out (not typed)
X
Image x, p + h
worked out (not typed)
Y
Image y, a·f(p) + k
worked out (not typed)
Formulas
q
=
p
2
1
=
1
2
X
=
p
+
h
4 = 1 + 3
Y
=
a
×
q
+
k
3 = 2 × 1 + 1
Assumptions
•
f(x − h) moves the graph h to the right; a negative h moves it left.
•
a·f(x) stretches every height by a (a negative a flips it), then + k moves it up by k.
•
So the point (p, f(p)) of f lands at (p + h, a·f(p) + k) on g.
Step-by-step
WE KNOW
a = 2, h = 3, k = 1, p = 1
FIND
f(p) = p² (q), image x, p + h (X), image y, a·f(p) + k (Y)
Step 1 · Find f(p) = p² (q)
Use q = p²
Put
p
into
f:
square
it.
q
=
1
2
q
=
1
Step 2 · Find image x, p + h (X)
Use X = p + h
Inside
the
brackets,
x
−
h
=
p
when
x
=
p
+
h
:
the
point
moves
h
across.
X
=
1
+
3
X
=
4
Step 3 · Find image y, a·f(p) + k (Y)
Use Y = a × q + k
Outside,
the
height
is
multiplied
by
a
,
then
k
is
added.
Y
=
2
×
1
+
1
Y
=
3
Check
Put the numbers back in. Both sides should match.
1
=
1
2
✓
4 = 1 + 3 ✓
3 = 2 × 1 + 1 ✓
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