Question

Use technology to find points and then graph the function y, equals, left parenthesis, one half, right parenthesis, start superscript, x, minus, 1, end superscript, minus, 10, commay=(
2
1
​
)
x−1
−10, following the instructions below.
Equation of asymptote:
y, equalsy=

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

Answer 1

To analyze the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \), we can start with identifying the asymptote and plotting points to graph the function.

### Step 1: Identify the Asymptote
The function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) approaches the horizontal line \( y = -10 \) as \( x \) increases, since the term \( \left(\frac{1}{2}\right)^{x-1} \) tends to zero.

**Equation of the asymptote:**
\[ y = -10 \]

### Step 2: Calculate Points
We will calculate at least five points where \( x \) is an integer.

1. **For \( x = 0 \)**:
\[
y = \left(\frac{1}{2}\right)^{0-1} - 10 = \left(\frac{1}{2}\right)^{-1} - 10 = 2 - 10 = -8
\]

2. **For \( x = 1 \)**:
\[
y = \left(\frac{1}{2}\right)^{1-1} - 10 = \left(\frac{1}{2}\right)^{0} - 10 = 1 - 10 = -9
\]

3. **For \( x = 2 \)**:
\[
y = \left(\frac{1}{2}\right)^{2-1} - 10 = \left(\frac{1}{2}\right)^{1} - 10 = \frac{1}{2} - 10 = -9.5
\]

4. **For \( x = 3 \)**:
\[
y = \left(\frac{1}{2}\right)^{3-1} - 10 = \left(\frac{1}{2}\right)^{2} - 10 = \frac{1}{4} - 10 = -9.75
\]

5. **For \( x = 4 \)**:
\[
y = \left(\frac{1}{2}\right)^{4-1} - 10 = \left(\frac{1}{2}\right)^{3} - 10 = \frac{1}{8} - 10 = -9.875
\]

### Points
Here are the points we've calculated:
- \( (0, -8) \)
- \( (1, -9) \)
- \( (2, -9.5) \)
- \( (3, -9.75) \)
- \( (4, -9.875) \)

### Summary of Points
We have five calculated points:
1. \( (0, -8) \)
2. \( (1, -9) \)
3. \( (2, -9.5) \)
4. \( (3, -9.75) \)
5. \( (4, -9.875) \)

### Graphing
To plot these points:
- **Draw the horizontal asymptote** at \( y = -10 \).
- Mark the calculated points on the graph, noting that as \( x \) increases, \( y \) approaches closer to -10 but never actually reaches it.

This will give us a clear view of the behavior of the function.