parent functions - houston independent school district€¦ · · 2014-10-20parent functions …...
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Parent Functions… 1) Linear:
2)Absolute Value: | |
3) Polynomial: or
4) Radical: √ or √
5) Rational:
or
6) Exponential:
7) Logarithmic:
8) Greatest Integer:
9) Sine/Cosine: or
Attributes of Functions
Transformations …
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Linear:
Constant Function
f(x) = c
D: (-∞,∞
R: 𝑐
Identity Function
f(x) = x
D: (-∞,∞
R: (-∞,∞
Linear Function
f(x) = mx + b
D: (-∞,∞
R: (-∞,∞
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Absolute Value: | |
| |
D: (-∞,∞
R: [0, ∞
Vertex: (h, k)
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Polynomial: or
𝒇 𝒙 𝒙𝒏 Where “n” is
positive and even
EX: 𝒚 𝒙𝟐
D: (-∞,∞
R: [0, ∞
Even function b/c
symmetric about y-axis
𝒇 𝒙 𝒙𝒏 Where “n” is
positive and odd
EX: 𝒚 𝒙𝟑
D: (-∞,∞
R: (-∞,∞
Odd function b/c
symmetric about origin
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Radical: √ or √
√
𝒇 𝒙 𝒙𝒏 Where “n” is
positive and even
EX: 𝒚 √𝒙
D: [0, ∞
R: [0, ∞
𝒇 𝒙 𝒙𝒏 Where “n” is
positive and odd
EX: 𝒚 √𝒙𝟑
D: (-∞,∞
R: (-∞,∞
Odd function b/c
symmetric about origin
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Rational:
or
𝒇 𝒙 𝟏
𝒙𝒏
Where “n” is
positive and odd
EX: 𝒚 𝟏
𝒙
𝒇 𝒙 𝟏
𝒙𝒏
Where “n” is positive
and even
EX: 𝒚 𝟏
𝒙𝟐
HA: y = 0
VA: x = 0
D: (-∞, 0 𝑈 0,∞
R: (-∞, 0 𝑈 0,∞
Odd function b/c
symmetric about origin
HA: y = 0
VA: x = 0
D: (-∞, 0 𝑈 0,∞
R: (0, ∞
Even function b/c
symmetric about y-axis
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Exponential:
𝒇 𝒙 𝒃𝒙
where b > 1
exponential growth
𝒇 𝒙 𝒃𝒙
where 0 < b < 1
exponential decay
(0,1) (0,1)
(1, b)
(1, b)
HA: y = 0
D: (-∞,∞
R: 0,∞
HA: y = 0
D: (-∞,∞
R: 0,∞
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Logarithmic:
logarithm base 10
log e x
natural logarithm
D: 0,∞
R: (-∞,∞
VA: x = 0
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Greatest Integer:
⌊ ⌋
D: (-∞,∞
R: … 2, 1, 0, 1, 2, … integers
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Sine/Cosine: or
Sine function
𝒇 𝒙 𝒔𝒊𝒏 𝒙
D: (-∞,∞
R: 1, 1
Odd function
b/c symmetric
about origin
Cosine function
𝒇 𝒙 𝒄𝒐𝒔 𝒙
D: (-∞,∞
R: 1, 1
Even function
b/c symmetric
about y-axis
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Attributes of Functions
Domain: x values
How far left and right
does the graph go?
(left, right)
D: (-∞,∞
Range: y values
How low and high
does the graph go?
(bottom, top)
R: (-∞, 2
Increasing: graph
goes up from left
to right
Decreasing: graph
goes down from
left to right
Constant: graph
remains horizontal
from left to right
Positive: the
part of the graph
above the x-axis
Negative: the
part of the graph
below the x-axis
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Transformations
Vertical
𝒚 𝒑 𝒙 𝒌 up k units
𝒚 𝒑 𝒙 𝒌 down k units
𝒚 𝒂 𝒑 𝒙 a > 1 - vertical stretch
- dilation by a
factor of “a”
0 < a < 1 - vertical compression
- dilation by a
factor of “a”
𝒚 𝒑 𝒙 reflection over
x-axis
𝒚 |𝒑 𝒙 | negative y values
reflect over x-axis
positive y values
remain the same
Horizontal
𝒚 𝒑 𝒙 𝒉 right h units
𝒚 𝒑 𝒙 𝒉 left h units
𝒚 𝒑 𝒃 𝒙 horizontal
dilation by a
factor of 1/b
𝒚 𝒑 𝒙 reflection over
y-axis
𝒚 𝒑 |𝒙| positive x values
remain the same
negative x values
reflect over x-axis