Question

Use technology to find points and then graph the function y, equals, 2, start superscript, x, minus, 4, end superscript, minus, 1, commay=2
x−4
−1, following the instructions below.
Equation of asymptote:
y, equalsy=

Plot at least four points with integer coordinates that fit on the axes below

Answer 1

To analyze the function \( y = 2^{(x - 4)} - 1 \) and find the points to plot, we can first identify the characteristics of this function and its asymptote.

### Step 1: Asymptote
The function \( y = 2^{(x - 4)} - 1 \) approaches the line \( y = -1 \) as \( x \) approaches negative infinity. Therefore, the equation of the asymptote is:
\[ y = -1 \]

### Step 2: Finding Points
To plot the function, we can calculate the \( y \) values for various integer \( x \) values. We'll choose a range of \( x \) values to see how the function behaves.

Let's evaluate the function at \( x = 0, 1, 2, 3, 4, 5, 6, 7 \):

- For \( x = 0 \):
\[
y = 2^{(0 - 4)} - 1 = 2^{-4} - 1 = \frac{1}{16} - 1 = \frac{1 - 16}{16} = -\frac{15}{16} \approx -0.9375
\]

- For \( x = 1 \):
\[
y = 2^{(1 - 4)} - 1 = 2^{-3} - 1 = \frac{1}{8} - 1 = \frac{1 - 8}{8} = -\frac{7}{8} = -0.875
\]

- For \( x = 2 \):
\[
y = 2^{(2 - 4)} - 1 = 2^{-2} - 1 = \frac{1}{4} - 1 = \frac{1 - 4}{4} = -\frac{3}{4} = -0.75
\]

- For \( x = 3 \):
\[
y = 2^{(3 - 4)} - 1 = 2^{-1} - 1 = \frac{1}{2} - 1 = \frac{1 - 2}{2} = -\frac{1}{2} = -0.5
\]

- For \( x = 4 \):
\[
y = 2^{(4 - 4)} - 1 = 2^{0} - 1 = 1 - 1 = 0
\]

- For \( x = 5 \):
\[
y = 2^{(5 - 4)} - 1 = 2^{1} - 1 = 2 - 1 = 1
\]

- For \( x = 6 \):
\[
y = 2^{(6 - 4)} - 1 = 2^{2} - 1 = 4 - 1 = 3
\]

- For \( x = 7 \):
\[
y = 2^{(7 - 4)} - 1 = 2^{3} - 1 = 8 - 1 = 7
\]

### Step 3: Integer Points
Let's focus on the integer coordinate points we found:
- \( (4, 0) \)
- \( (5, 1) \)
- \( (6, 3) \)
- \( (7, 7) \)

### Results
Here are the integer coordinates we can plot:
1. \( (4, 0) \)
2. \( (5, 1) \)
3. \( (6, 3) \)
4. \( (7, 7) \)

### Graphing the Function
On a coordinate plane, we can mark these points. The asymptote \( y = -1 \) will also be represented as a dashed horizontal line across the graph at \( y = -1 \).

Here's a simple representation of the function:
- The curve rises steeply after \( x = 4 \).
- It approaches the asymptote (horizontal line at \( y = -1 \)) for smaller values of \( x \).

### Final Summary
- **Equation of Asymptote:** \( y = -1 \)
- **Integer Points to Plot:**
- \( (4, 0) \)
- \( (5, 1) \)
- \( (6, 3) \)
- \( (7, 7) \)

You can plot these points and sketch the curve for a complete view of the function behavior!