Exponential Functions Notes and Worksheets - Lindsay Bowden - Free Printable

Let’s solve each problem step by step. We’ll go one at a time, carefully checking our work.

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Problem 1: Use the graph to find characteristics of the function.

We’re given a graph that looks like an exponential growth curve — it starts low on the left and rises sharply to the right. It passes through (0,1) and seems to approach y=0 as x goes to negative infinity.

- Growth or decay? → Since the graph is going up as x increases, this is growth.
- Asymptote at? → The graph gets closer and closer to y = 0 but never touches it → asymptote at y = 0.
- Domain: All real numbers → (-∞, ∞)
- Range: Since it’s above y=0 and growing → (0, ∞)
- y-intercept: Where x=0 → from graph, it’s at (0,1) → 1

✔ Checked: Graph matches standard exponential growth f(x)=a·b^x with b>1, passing through (0,1).

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Problem 2: Use the graph to find characteristics of f(x) = 3·(1/2)^x + 4

Graph shows a decreasing curve approaching y=4 from above. At x=0, y=7. As x→∞, y→4. As x→-∞, y→∞.

This is exponential decay because base (1/2) < 1.

- Growth or decay? → Base is 1/2 (<1), so decay
- Asymptote at? → Horizontal asymptote is y = 4 (the "+4" shifts it up)
- Domain: Always all real numbers for exponentials → (-∞, ∞)
- Range: Since it decays toward y=4 from above, range is (4, ∞)
- y-intercept: Plug in x=0:
f(0) = 3·(1/2)^0 + 4 = 3·1 + 4 = 7

✔ Checked: Matches graph — starts high, decreases toward y=4, crosses y-axis at 7.

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Problem 3: Use table to graph f(x) = 4^(x−2) − 3

First, fill in the table:

| x | f(x) = 4^(x−2) − 3 |
|----|-----------------------------|
| -2 | 4^(-4) − 3 = 1/256 − 3 ≈ -2.996 → round to -3.0 (or keep exact if needed, but for graphing approx ok) |
| -1 | 4^(-3) − 3 = 1/64 − 3 ≈ -2.984 → -3.0 |
| 0 | 4^(-2) − 3 = 1/16 − 3 = -2.9375 → -2.9 |
| 1 | 4^(-1) − 3 = 1/4 − 3 = -2.75 → -2.8 |
| 2 | 4^(0) − 3 = 1 − 3 = -2 |
| 3 | 4^(1) − 3 = 4 − 3 = 1 |

Actually, let’s compute exactly for accuracy:

- x=-2: 4^(-4) = 1/256 → 1/256 - 3 = -767/256 ≈ -2.996 → we can write as fraction or decimal. For graphing, use decimals rounded to 1 place unless specified.

But since the problem says “use the table”, maybe they expect exact values? Let’s do both.

Better to compute precisely:

f(-2) = 4^(-4) - 3 = 1/256 - 3 = (1 - 768)/256 = -767/256
f(-1) = 4^(-3) - 3 = 1/64 - 3 = (1 - 192)/64 = -191/64
f(0) = 4^(-2) - 3 = 1/16 - 3 = (1 - 48)/16 = -47/16
f(1) = 4^(-1) - 3 = 1/4 - 3 = -11/4
f(2) = 4^0 - 3 = 1 - 3 = -2
f(3) = 4^1 - 3 = 4 - 3 = 1

For graphing purposes, we can use approximate decimals:

| x | f(x) |
|----|----------|
| -2 | -3.0 |
| -1 | -3.0 |
| 0 | -2.9 |
| 1 | -2.8 |
| 2 | -2.0 |
| 3 | 1.0 |

Now, characteristics:

- Growth or decay? → Base is 4 > 1 → growth
- Asymptote at? → Exponential part 4^(x−2) approaches 0 as x→-∞, so f(x) → -3 → asymptote at y = -3
- Domain: All real numbers → (-∞, ∞)
- Range: Since it grows from near -3 upward → (-3, ∞)
- y-intercept: When x=0 → f(0) = 4^(-2) - 3 = 1/16 - 3 = -47/16 = -2.9375 → or -2.9 if rounding

✔ Checked: Function is shifted right 2 and down 3 from parent 4^x. So yes, asymptote y=-3, growth, etc.

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Problem 4: Use table to graph f(x) = 2·(1/2)^x − 3

Fill table:

| x | f(x) = 2·(1/2)^x − 3 |
|----|------------------------------|
| -2 | 2·(1/2)^(-2) − 3 = 2·4 − 3 = 8 − 3 = 5 |
| -1 | 2·(1/2)^(-1) − 3 = 2·2 − 3 = 4 − 3 = 1 |
| 0 | 2·(1/2)^0 − 3 = 2·1 − 3 = 2 − 3 = -1 |
| 1 | 2·(1/2)^1 − 3 = 2·0.5 − 3 = 1 − 3 = -2 |
| 2 | 2·(1/2)^2 − 3 = 2·0.25 − 3 = 0.5 − 3 = -2.5 |
| 3 | 2·(1/2)^3 − 3 = 2·0.125 − 3 = 0.25 − 3 = -2.75 |

So table:

| x | f(x) |
|----|--------|
| -2 | 5 |
| -1 | 1 |
| 0 | -1 |
| 1 | -2 |
| 2 | -2.5 |
| 3 | -2.75 |

Characteristics:

- Growth or decay? → Base is 1/2 < 1 → decay
- Asymptote at? → As x→∞, (1/2)^x → 0, so f(x) → -3 → asymptote at y = -3
- Domain: All real numbers → (-∞, ∞)
- Range: Since it decays toward y=-3 from above (starts high and goes down to -3), range is (-3, ∞)
- y-intercept: x=0 → f(0) = -1 → -1

✔ Checked: Matches behavior — starts high when x negative, decreases toward y=-3.

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Final Answers compiled:

Problem 1:
- Growth or decay? → growth
- Asymptote at? → y = 0
- Domain: (-∞, ∞)
- Range: (0, ∞)
- y-intercept: 1

Problem 2:
- Growth or decay? → decay
- Asymptote at? → y = 4
- Domain: (-∞, ∞)
- Range: (4, ∞)
- y-intercept: 7

Problem 3:
Table filled:
x | f(x)
-2 | -767/256 ≈ -3.0
-1 | -191/64 ≈ -3.0
0 | -47/16 = -2.9375
1 | -11/4 = -2.75
2 | -2
3 | 1

Characteristics:
- Growth or decay? → growth
- Asymptote at? → y = -3
- Domain: (-∞, ∞)
- Range: (-3, ∞)
- y-intercept: -47/16 or -2.9375

Problem 4:
Table filled:
x | f(x)
-2 | 5
-1 | 1
0 | -1
1 | -2
2 | -2.5
3 | -2.75

Characteristics:
- Growth or decay? → decay
- Asymptote at? → y = -3
- Domain: (-∞, ∞)
- Range: (-3, ∞)
- y-intercept: -1

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Final Answer:

Problem 1:
- Growth or decay? → growth
- Asymptote at? → y = 0
- Domain: (-∞, ∞)
- Range: (0, ∞)
- y-intercept: 1

Problem 2:
- Growth or decay? → decay
- Asymptote at? → y = 4
- Domain: (-∞, ∞)
- Range: (4, ∞)
- y-intercept: 7

Problem 3:
Table:
x | f(x)
-2 | -767/256 (≈ -3.0)
-1 | -191/64 (≈ -3.0)
0 | -47/16 (-2.9375)
1 | -11/4 (-2.75)
2 | -2
3 | 1

Characteristics:
- Growth or decay? → growth
- Asymptote at? → y = -3
- Domain: (-∞, ∞)
- Range: (-3, ∞)
- y-intercept: -47/16 (or -2.9375)

Problem 4:
Table:
x | f(x)
-2 | 5
-1 | 1
0 | -1
1 | -2
2 | -2.5
3 | -2.75

Characteristics:
- Growth or decay? → decay
- Asymptote at? → y = -3
- Domain: (-∞, ∞)
- Range: (-3, ∞)
- y-intercept: -1