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M
Exponential Functions
course notation
book notation
• ‘a’ is called the initial value.
• ‘a’ determines vertical stretch/compression.
• if a < 0 then reflection across the x-axis.
• ‘c’ is called the base or growth factor.
• ‘b’ determines horizontal stretch/compression.
• if b < 0 then reflection across the y-axis.
• ‘h’ determines the horizontal shift left or right.
• ‘k’ determines the vertical shift down or up.
• horizontal asymptote at y = k
• ‘C’ is called the initial value.
• ‘C’ determines vertical stretch/compression.
• if C < 0 then reflection across the x-axis.
• ‘a’ is called the base or growth factor.
• ‘b’ determines horizontal stretch/compression.
• if b < 0 then reflection across the y-axis.
• ‘h’ determines the horizontal shift left or right.
• ‘k’ determines the vertical shift down or up.
• horizontal asymptote at y = k
y=
c-x
y=
cx
using finite differences to classify function type
x
y
(-1,c)
y = 2x
-3 -2 -1 0 1 2 3
-6 -4 -2 0 2 4 6
V V V V V V
linear:
(1,c)
+2
(0,1)
+2
V
+0
(0,-1)
(-1,-c)
x
y
HA: y = 0
y = -cx
V
+0
+2
V
+0
+2
V
+0
+2
V
+0
y = x2
-3 -2 -1 0 1 2 3
9 4 1 0 1 4 9
V V V V V V
quadratic:
(1,-c)
y = -c-x
+2
-5
V
+2
-3
V
+0
V
+2
-1
V
+0
+1
V
+2
V
+0
+3
V
+2
V
+0
+5
V
+2
y = 2x
-3 -2 -1 0 1 2 3
1 2 4 8
V V V V V V
exponential:
x
y
+1
V
V
V
+2
V
+1
ratio of consecutive outputs (y-values)
is the base (growth factor)
+4
V
+2
graphing using transformations
x
-2
-1
0
1
2
y
1
3
9
HA: y = 0
x y
-13
-9
-5 -8
-1 -12
3 -24
HA: y = -6
• reflected across the x-axis
• vertical stretch by a factor of 2
• horizontal stretch by a factor of 4
• horizontal shift 5 units to the left
• vertical shift 6 units down
solve for the exponent using equal bases