Coordinate Algebra
Exponential Functions Day 2 Notes
Date: _______
COMPLETED
Exponential functions, just as any function, can be transformed by addition, subtraction, and multiplication.
When this occurs in a problem the original function will shift left or right, up or down, and reflect.
Vertical Shifts
A function can travel along the y-axis by adding or subtracting within the function.
Ex1) f(x) = 2x + 3
The adding 3 at the end of the function will shift the graph up three on the y-axis
Original Function VS. Shifted Function
Shifted Function
Original Function
Vertical Shifts
change the
asymptote!
Ex2) What would happen to this function: f(x) = 2x – 3
This function will shift down 3 on the y-axis.
Horizontal Shifts
A function can travel along the x-axis by adding or subtracting within the function as well.
Ex3) f(x) = 2x+3
The adding three in the exponent will shift the graph along the x-axis to the left.
Ex4) f(x) = 2x – 3
The subtracting three in the exponent will shift the graph along the x-axis to the right.
Reflections
When a function is multiplied by a negative 1, the function will be reflected over the x-axis. This turns an
exponential decay function into a reflected exponential decay and an exponential growth function into a
reflected exponential growth. It is important to remember that these functions are still growth or decay; the
reflection does not change that!
1 𝑥
Ex5) f(x) = −(2)x
Ex6) f(x) = −(2)
This is an exponential growth function
reflected over the x-axis.
This is an exponential decay function
reflected over the x-axis.
These transformations affect the characteristics such as Range, End Behavior, Increasing/Decreasing, and
Asymptote. They will also affect the x-values we choose for the table if the graph is shifted horizontally. If the
function has been shifted left or right by addition in the exponent then you need to adjust your table values.
Instead of using the standard (-1, 0, 1) values you will use the adjusted values as seen in the example below.
Ex7) f(x) = 2x+3
Instead of using -1, 0, 1
subtract 3 to all these values
and use those in the table
-1 − 3 = -4
0 − 3 = -3
1 − 3 = -2
x
f(x)
-4 ½ or 0.5
-3
1
-2
2
y=0
*We do this to make the table values easier points to graph.
Circle One:
Exponential Growth
Exponential Decay
Reflected Exponential Growth
Reflected Exponential Decay
Domain: _____ ℝ _________ Range: ____ y > 0 ________ Asymptote: _____ y = 0 _______
Increasing or Decreasing
End Behavior: x  ∞, y  __ ∞ ___; x  −∞, y  __ 0 ____
𝟕
Average Rate of Change for -3 ≤ x ≤ 0: ____ 𝟑 𝒐𝒓 𝟐. 𝟑𝟑 _______
(-3, 1) & (0, 8)
𝟖−𝟏
𝟕
= 𝟑 ≈ 𝟐. 𝟑𝟑
𝟎−−𝟑
3 𝑥
Ex8) f(x) = (2) + 4
x
-1
𝟏𝟒
𝟑
or 4.67
0
1
y=4
f(x)
5
𝟏𝟏
𝟐
or 5.5
Circle One:
Exponential Growth
Exponential Decay
Reflected Exponential Growth
Reflected Exponential Decay
Domain: _____ ℝ _________ Range: ____ y > 4 ________ Asymptote: _____ y = 4 _______
Increasing or Decreasing
End Behavior: x  ∞, y  __ ∞ ___; x  −∞, y  __ 4 ____
𝟏
Average Rate of Change for 0 ≤ x ≤ 1: _____ 𝟐 𝒐𝒓 𝟎. 𝟓 __________
(0, 5) & (1, 5.5)
𝟓.𝟓−𝟓
𝟎.𝟓
= 𝟏 = 𝟎. 𝟓
𝟏−𝟎
Ex9) f(x) = −(2)𝑥−3
Instead of using -1, 0, 1
add 3 to all these values
and use those in the table
-1 + 3 = 2
0+3=3
1+3=4
Circle One:
Exponential Growth
x
f(x)
2 −½ or −0.5
3
−1
4
−2
Exponential Decay
Reflected Exponential Growth
y=0
Reflected Exponential Decay
Domain: _____ ℝ _________ Range: ____ y < 0 ________ Asymptote: _____ y = 0 _______
Increasing or Decreasing
End Behavior: x  ∞, y  __ −∞ ___; x  −∞, y  __ 0 ____
Average Rate of Change for 3 ≤ x ≤ 4: ______ -1 _________
(3, -1) & (4, -2)
−𝟐−−𝟏
−𝟏
= 𝟏 = −𝟏
𝟒−𝟑
2 𝑥−2
Ex10) f(x) = (5)
−3
Instead of using -1, 0, 1
add 2 to all these values
and use those in the table
x
f(x)
1 −½ or −0.5
−2
2
-1 + 2 = 1
0+2=2
1+2=3
3
Circle One:
Exponential Growth
−
𝟏𝟑
𝟓
y = -3
or -2.6
Exponential Decay
Reflected Exponential Growth
Reflected Exponential Decay
Domain: _____ ℝ _________ Range: ____ y > -3 ________ Asymptote: _____ y = -3 _______
Increasing or Decreasing
End Behavior: x  ∞, y  __ −3 ___; x  −∞, y  __ ∞ ____
𝟑
Average Rate of Change for 1 ≤ x ≤ 2: ______ − 𝟐 or −1.5 _________
(1, -0.5) & (2, -2)
−𝟐−−𝟎.𝟓
−𝟏.𝟓
= 𝟏 = −𝟏. 𝟓
𝟐−𝟏
1 𝑥
Ex11) f(x) = −(3) + 2
x
f(x)
-1
−1
0
1
1
𝟓
𝟑
y=2
or 1.67
Circle One:
Exponential Growth
Exponential Decay
Reflected Exponential Growth
Reflected Exponential Decay
Domain: _____ ℝ _________ Range: ____ y < 2 ________ Asymptote: _____ y = 2 _______
Increasing or Decreasing
End Behavior: x  ∞, y  __ 2 ___; x  −∞, y  __ −∞ ____
Average Rate of Change for -2 ≤ x ≤ 0: ______ 4 _________
(-2, -7) & (0, 1)
𝟏−−𝟕
𝟖
=𝟐=𝟒
𝟎−−𝟐