ACE-Learning Lesson Brief
Graphs of Power Functions, y = axn where n = –2, –1, 0, 1, 2, 3
Introduction:
Power (n)
y = axn
0
y=a
Function
Type
Constant
Graph
Observation
If a is
positive : line lies above x-axis
negative : line lies below x-axis
1
y = ax
Linear
If a is
positive : line rises from left to
right
negative : line falls from left to
right
2
y = ax2
Quadratic
If a is
positive : Curve is ∪-shaped
negative : Curve is ∩-shaped
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ACE-Learning Lesson Brief
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y = ax3
Cubic
If a is
positive : cubic “S” goes up
and to the right
negative : cubic “S” slopes
down and to the
left
–1
–2
y
a
x
Reciprocal
or
Rectangular
Hyperbolic
y
a
x2
Square
Reciprocal
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1. The function is undefined
at x = 0.
2. The graph does not touch
the axes.
3. The graph is not
continuous and appears to
have two separate parts
but it must be regarded as
a single graph
4. The two parts are mirror
images of each other.
5. The lines of symmetry are
y = x and y = –x.
6. x tends to 0 as y increases
or decreases infinitely.
7. y tends to 0 as x increases
or decreases infinitely.
1. The function is
undefined at x = 0.
2. The graph does not touch
the axes.
3. The graph is not
continuous and appears
to have two separate
parts but it must be
regarded as a single
graph
4. The two parts are mirror
images of each other.
5. The line of symmetry is
the y-axis.
6. If a is positive:
(a) y is always positive;
so the curve lies
above the axis
(b) x tends to 0 as y
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ACE-Learning Lesson Brief
increases infinitely
7. If a is negative:
(a) y is always negative;
so the curve lies
below the axis
(b) x tends to 0 as y
decreases infinitely
8. y tends to 0 as x
increases or decreases
infinitely.
Practice Questions:
1.
Draw the graph of y = 3x – 4 for 0 ≤ x ≤ 3.
2.
Draw the graph of y = x2 + 2x – 3 for –4 ≤ x ≤ 2
3.
Draw the graph of y = x3 – 2x2 – 3x for –2 ≤ x ≤ 4.
4.
Draw the graph of y  x 
5.
Draw the graph of y 
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for 0.5 ≤ x ≤ 5.
x
1
 x for 0.5 ≤ x ≤ 3.
x2
3