Question

Use technology to find points and then graph the function y, equals, left parenthesis, one half, right parenthesis, start superscript, x, plus, 2, end superscript, minus, 9, commay=(
2
1
​
)
x+2
−9, 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 \( y = \left( \frac{1}{2} \right)^{x + 2} - 9 \), let's find the horizontal asymptote and plot points.

### Step 1: Find the Equation of the Asymptote

As \( x \) approaches infinity, \( \left( \frac{1}{2} \right)^{x + 2} \) approaches 0. Therefore, the horizontal asymptote for the function is:

\[
y = -9
\]

### Step 2: Calculate Points

Let's substitute five integer values for \( x \) to calculate \( y \):
1. **For \( x = -5 \):**
\[
y = \left( \frac{1}{2} \right)^{-5 + 2} - 9 = \left( \frac{1}{2} \right)^{-3} - 9 = 8 - 9 = -1
\]
Point: \((-5, -1)\)

2. **For \( x = -4 \):**
\[
y = \left( \frac{1}{2} \right)^{-4 + 2} - 9 = \left( \frac{1}{2} \right)^{-2} - 9 = 4 - 9 = -5
\]
Point: \((-4, -5)\)

3. **For \( x = -3 \):**
\[
y = \left( \frac{1}{2} \right)^{-3 + 2} - 9 = \left( \frac{1}{2} \right)^{-1} - 9 = 2 - 9 = -7
\]
Point: \((-3, -7)\)

4. **For \( x = -2 \):**
\[
y = \left( \frac{1}{2} \right)^{-2 + 2} - 9 = \left( \frac{1}{2} \right)^{0} - 9 = 1 - 9 = -8
\]
Point: \((-2, -8)\)

5. **For \( x = -1 \):**
\[
y = \left( \frac{1}{2} \right)^{-1 + 2} - 9 = \left( \frac{1}{2} \right)^{1} - 9 = \frac{1}{2} - 9 = -\frac{17}{2} \approx -8.5
\]
Point: \((-1, -8.5)\)

6. **For \( x = 0 \):**
\[
y = \left( \frac{1}{2} \right)^{0 + 2} - 9 = \left( \frac{1}{2} \right)^{2} - 9 = \frac{1}{4} - 9 = -\frac{35}{4} \approx -8.75
\]
Point: \((0, -8.75)\)

7. **For \( x = 1 \):**
\[
y = \left( \frac{1}{2} \right)^{1 + 2} - 9 = \left( \frac{1}{2} \right)^{3} - 9 = \frac{1}{8} - 9 = -\frac{71}{8} \approx -8.875
\]
Point: \((1, -8.875)\)

### Step 3: Summary of Points

Here’s the list of points we obtained:
- \((-5, -1)\)
- \((-4, -5)\)
- \((-3, -7)\)
- \((-2, -8)\)
- \((-1, -8.5)\)
- \( (0, -8.75) \)
- \( (1, -8.875) \)

### Step 4: Graphing the Function

To graph the points:
- Start by plotting the asymptote line at \( y = -9 \).
- Then plot each of the calculated points on a coordinate grid.
- Connect the points smoothly to show the curve approaching the asymptote as \( x \) increases.

Here's a rough description of how it will look:
- The graph will start above the asymptote and drop towards it as \( x \) increases, flattening out near \( y = -9 \).

Consider using graphing software or a calculator to visualize this. If you'd like to graph it manually, just ensure that your points are correctly plotted and that you draw the curve approaching the horizontal asymptote at \( y = -9 \).