Question: User: GENERAL INSTRUCTIONS For each item, do the following in sequence:
(a) Fill up the table of values. Note that the first one is already done for you.
(b) Sketch the all the given exponential functions in one Cartesian Plane.
(c) Describe the graphs. Write your observation after the graph.
1. Given: y = 2^x ; y = 3^x ; y = 4^x
x 63 2 1 0 1 2 3
y = 2^x 1/8 or 0.125 1/4 or 0.25 1/2 or 0.5 1 2 4 8
y = 3^x
y = 4^x
1. **State the problem:**
We are given three exponential functions: $$y=2^x$$, $$y=3^x$$, and $$y=4^x$$.
We need to:
- (a) Fill the table of values for $$y=3^x$$ and $$y=4^x$$ for $$x = -3, -2, -1, 0, 1, 2, 3$$.
- (b) Sketch all three functions on the same Cartesian plane.
- (c) Describe the graphs and write observations.
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2. **Fill the table values:**
Recall that for exponential functions $$y = a^x$$, when $$x$$ is negative, $$y = \frac{1}{a^{-x}}$$.
Calculate for $$y=3^x$$:
- $$x=-3: y=3^{-3} = \frac{1}{3^3} = \frac{1}{27} \approx 0.037$$
- $$x=-2: y=3^{-2} = \frac{1}{3^2} = \frac{1}{9} \approx 0.111$$
- $$x=-1: y=3^{-1} = \frac{1}{3} \approx 0.333$$
- $$x=0: y=3^0 = 1$$
- $$x=1: y=3^1 = 3$$
- $$x=2: y=3^2 = 9$$
- $$x=3: y=3^3 = 27$$
Calculate for $$y=4^x$$:
- $$x=-3: y=4^{-3} = \frac{1}{4^3} = \frac{1}{64} \approx 0.0156$$
- $$x=-2: y=4^{-2} = \frac{1}{4^2} = \frac{1}{16} = 0.0625$$
- $$x=-1: y=4^{-1} = \frac{1}{4} = 0.25$$
- $$x=0: y=4^0 = 1$$
- $$x=1: y=4^1 = 4$$
- $$x=2: y=4^2 = 16$$
- $$x=3: y=4^3 = 64$$
**Completed table:**
| x | $$y=2^x$$ | $$y=3^x$$ | $$y=4^x$$ |
|----|-----------|-----------|-----------|
| -3 | 0.125 | 0.037 | 0.0156 |
| -2 | 0.25 | 0.111 | 0.0625 |
| -1 | 0.5 | 0.333 | 0.25 |
| 0 | 1 | 1 | 1 |
| 1 | 2 | 3 | 4 |
| 2 | 4 | 9 | 16 |
| 3 | 8 | 27 | 64 |
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3. **Sketch the graphs:**
- All three graphs pass through the point $$(0,1)$$ because $$a^0=1$$ for any $$a>0$$.
- For negative $$x$$, the values approach zero but never reach it (asymptote at $$y=0$$).
- For positive $$x$$, the functions grow exponentially, with $$y=4^x$$ growing fastest, then $$y=3^x$$, then $$y=2^x$$.
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4. **Observations:**
- All graphs are increasing exponential functions.
- The base determines the growth rate: larger base means faster growth.
- The graphs approach the x-axis (y=0) as $$x \to -\infty$$ but never touch it.
- The point $$(0,1)$$ is common to all.
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**Final answer:**
The table is filled as above.
The graphs show exponential growth with different rates depending on the base.
The functions are all increasing, continuous, and have horizontal asymptote $$y=0$$.