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

Answer 1

To analyze the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) and find its asymptote and points for graphing, we can follow these steps:

### Step 1: Find the equation of the asymptote
The function is an exponential decay function, which typically has a horizontal asymptote. As \( x \) approaches infinity, \( \left(\frac{1}{2}\right)^{x-1} \) approaches 0, so we can see if there's a vertical shift affecting the asymptote.

Thus, let's find the horizontal asymptote:
\[
\lim_{{x \to \infty}} \left(\frac{1}{2}\right)^{x-1} - 10 = 0 - 10 = -10
\]
The equation of the horizontal asymptote is:
\[
y = -10
\]

### Step 2: Plot at least five points with integer coordinates
We can calculate the function values for integer values of \( x \).

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

2. **When \( x = 2 \)**:
\[
y = \left(\frac{1}{2}\right)^{2-1} - 10 = \frac{1}{2} - 10 = -9.5
\]
Point: \( (2, -9.5) \) (not an integer, skip)

3. **When \( x = 3 \)**:
\[
y = \left(\frac{1}{2}\right)^{3-1} - 10 = \frac{1}{4} - 10 = -9.75
\]
Point: \( (3, -9.75) \) (not an integer, skip)

4. **When \( x = 4 \)**:
\[
y = \left(\frac{1}{2}\right)^{4-1} - 10 = \frac{1}{8} - 10 = -9.875
\]
Point: \( (4, -9.875) \) (not an integer, skip)

5. **When \( x = 5 \)**:
\[
y = \left(\frac{1}{2}\right)^{5-1} - 10 = \frac{1}{16} - 10 = -9.9375
\]
Point: \( (5, -9.9375) \) (not an integer, skip)

6. **When \( x = 6 \)**:
\[
y = \left(\frac{1}{2}\right)^{6-1} - 10 = \frac{1}{32} - 10 \approx -9.96875
\]
Point: \( (6, -9.96875) \) (not an integer, skip)

7. **When \( x = 7 \)**:
\[
y = \left(\frac{1}{2}\right)^{7-1} - 10 = \frac{1}{64} - 10 \approx -9.984375
\]
Point: \( (7, -9.984375) \) (not an integer, skip)

We can make calculations for negative \( x \) values:

1. **When \( x = 0 \)**:
\[
y = \left(\frac{1}{2}\right)^{0-1} - 10 = \frac{1}{0.5} - 10 = 2 - 10 = -8
\]
Point: \( (0, -8) \)

2. **When \( x = -1 \)**:
\[
y = \left(\frac{1}{2}\right)^{-1-1} - 10 = \left(\frac{1}{2}\right)^{-2} - 10 = 4 - 10 = -6
\]
Point: \( (-1, -6) \)

3. **When \( x = -2 \)**:
\[
y = \left(\frac{1}{2}\right)^{-2-1} - 10 = \left(\frac{1}{2}\right)^{-3} - 10 = 8 - 10 = -2
\]
Point: \( (-2, -2) \)

4. **When \( x = -3 \)**:
\[
y = \left(\frac{1}{2}\right)^{-3-1} - 10 = \left(\frac{1}{2}\right)^{-4} - 10 = 16 - 10 = 6
\]
Point: \( (-3, 6) \)

5. **When \( x = -4 \)**:
\[
y = \left(\frac{1}{2}\right)^{-4-1} - 10 = \left(\frac{1}{2}\right)^{-5} - 10 = 32 - 10 = 22
\]
Point: \( (-4, 22) \)

### Summary of Points
Here are the points with integer coordinates we found:
1. \( (0, -8) \)
2. \( (-1, -6) \)
3. \( (-2, -2) \)
4. \( (-3, 6) \)
5. \( (-4, 22) \)

### Final Result
The equation of the horizontal asymptote is:
\[
y = -10
\]

The points to plot on the graph of the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) are:
1. \( (0, -8) \)
2. \( (-1, -6) \)
3. \( (-2, -2) \)
4. \( (-3, 6) \)
5. \( (-4, 22) \)

You can graph the function based on these points and observe the horizontal asymptote at \( y = -10 \).