Worksheets on transformations of exponential functions with step-by-step instructions and graphing exercises.

Let’s solve the problems shown in the worksheet step by step. We’ll focus on the two fully worked examples at the bottom right of the page — they show how to analyze and graph transformed exponential functions.

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We are given two functions:

Problem 1:
g(x) = 4^x - 3

Problem 2:
g(x) = 3*(1/2)^x

We need to find for each:
- Parent function
- Table of values (for x = -2, -1, 0, 1, 2)
- Y-intercept
- Asymptote
- Whether it’s growth or decay
- Transformations from parent
- Domain and range

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## ✔ Problem 1: g(x) = 4^x - 3

Step 1: Identify the parent function

The parent function is the basic exponential form without any transformations.
→ Parent: y = 4^x

Step 2: Create a table of values for the parent function first, then apply transformation

Parent: y = 4^x

| x | 4^x |
|----|------------|
| -2 | 4^(-2) = 1/16 |
| -1 | 4^(-1) = 1/4 |
| 0 | 4^0 = 1 |
| 1 | 4^1 = 4 |
| 2 | 4^2 = 16 |

Now subtract 3 from each y-value to get g(x) = 4^x - 3:

| x | g(x) = 4^x - 3 |
|----|--------------------|
| -2 | 1/16 - 3 = -47/16 ≈ -2.9375 → but worksheet says -47/16? Wait — let’s check what’s written.

Wait — looking at the image, the table shows:

For g(x)=4^x -3:

x | y
--|--
-2 | -47/16? But that doesn’t match 1/16 - 3 = -47/16? Actually yes: 1/16 - 48/16 = -47/16 ✔️

But in the image, it's written as “-47/16” for x=-2? Let me recalculate carefully:

Actually, 4^(-2) = 1/(4^2) = 1/16
Then 1/16 - 3 = 1/16 - 48/16 = -47/16

Similarly:

x = -1: 4^(-1) = 1/4 → 1/4 - 3 = 1/4 - 12/4 = -11/4

x = 0: 1 - 3 = -2

x = 1: 4 - 3 = 1

x = 2: 16 - 3 = 13

So table should be:

| x | y |
|----|-----------|
| -2 | -47/16 |
| -1 | -11/4 |
| 0 | -2 |
| 1 | 1 |
| 2 | 13 |

✔ Matches the handwritten answer in the image.

Step 3: Y-intercept

Y-intercept occurs when x = 0 → we already calculated: y = -2
→ So point: (0, -2)

Step 4: Asymptote

Original parent function y = 4^x has horizontal asymptote at y = 0.
We shifted down by 3 → new asymptote: y = -3

Step 5: Growth or Decay?

Base = 4 > 1 → so original is growth. Shifting down doesn’t change that.
→ Still Growth

Step 6: Transformation(s)

From y = 4^x to y = 4^x - 3 → vertical shift down 3 units

Step 7: Domain and Range

Domain: All real numbers → (-∞, ∞)
Range: Since asymptote is y = -3 and graph goes up forever → y > -3 or (-3, ∞)

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## ✔ Problem 2: g(x) = 3*(1/2)^x

Step 1: Parent function

Basic form: y = (1/2)^x
→ Parent: y = (1/2)^x

Step 2: Table of values

First compute parent: y = (1/2)^x

| x | (1/2)^x |
|----|-------------|
| -2 | (1/2)^(-2) = 4 |
| -1 | (1/2)^(-1) = 2 |
| 0 | 1 |
| 1 | 1/2 |
| 2 | 1/4 |

Now multiply each by 3:

g(x) = 3 * (1/2)^x

| x | g(x) |
|----|---------------|
| -2 | 3*4 = 12 |
| -1 | 3*2 = 6 |
| 0 | 3*1 = 3 |
| 1 | 3*(1/2) = 3/2 |
| 2 | 3*(1/4) = 3/4 |

✔ Matches the handwritten table.

Step 3: Y-intercept

At x=0 → y=3 → (0, 3)

Step 4: Asymptote

Parent y=(1/2)^x has asymptote y=0. Multiplying by 3 doesn’t move the asymptote → still y = 0

Step 5: Growth or Decay?

Base = 1/2 < 1 → this is decay. Even though we multiplied by 3, the base determines growth/decay.
→ Decay

Step 6: Transformation(s)

From y = (1/2)^x to y = 3*(1/2)^x → vertical stretch by factor of 3

Step 7: Domain and Range

Domain: All real numbers → (-∞, ∞)
Range: Since asymptote is y=0 and all outputs are positive → y > 0 or (0, ∞)

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## Final Answers Summary:

For g(x) = 4^x - 3:

- Parent Function: y = 4^x
- Table:
x: -2 → y: -47/16
x: -1 → y: -11/4
x: 0 → y: -2
x: 1 → y: 1
x: 2 → y: 13
- Y-Intercept: (0, -2)
- Asymptote: y = -3
- Growth or Decay: Growth
- Transformation: Vertical translation down 3 units
- Domain: All real numbers
- Range: y > -3 or (-3, ∞)

For g(x) = 3*(1/2)^x:

- Parent Function: y = (1/2)^x
- Table:
x: -2 → y: 12
x: -1 → y: 6
x: 0 → y: 3
x: 1 → y: 3/2
x: 2 → y: 3/4
- Y-Intercept: (0, 3)
- Asymptote: y = 0
- Growth or Decay: Decay
- Transformation: Vertical stretch by a factor of 3
- Domain: All real numbers
- Range: y > 0 or (0, ∞)

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Final Answer:
For g(x) = 4^x - 3:
Parent: y = 4^x; Table: [(-2, -47/16), (-1, -11/4), (0, -2), (1, 1), (2, 13)]; Y-int: (0, -2); Asymptote: y = -3; Growth; Transformation: Down 3; Domain: all reals; Range: y > -3

For g(x) = 3*(1/2)^x:
Parent: y = (1/2)^x; Table: [(-2, 12), (-1, 6), (0, 3), (1, 3/2), (2, 3/4)]; Y-int: (0, 3); Asymptote: y = 0; Decay; Transformation: Vertical stretch by 3; Domain: all reals; Range: y > 0