Question

Use technology to find points and then graph the function y, equals, 2, start superscript, x, minus, 2, end superscript, minus, 5, commay=2
x−2
−5, following the instructions below.
Equation of asymptote:
y, equalsy=

Plot at least four points with integer coordinates

Answers

Answered by GPT-4o mini
To analyze the function \( y = 2^{(x - 2)} - 5 \) and find its features including the asymptote and integer coordinate points, we can follow these steps:

### Step 1: Determine the Equation of Asymptote
The function \( y = 2^{(x - 2)} - 5 \) is an exponential function. The horizontal asymptote occurs as \( x \to \infty \) or \( x \to -\infty \).

To find the asymptote, we can observe that as \( x \to -\infty \), \( 2^{(x - 2)} \) approaches 0. Therefore, the asymptote is:

\[
y = -5
\]

### Step 2: Calculate Points with Integer Coordinates
Now we can calculate points by choosing integer values of \( x \):

1. **For \( x = 0 \)**:
\[
y = 2^{(0 - 2)} - 5 = 2^{-2} - 5 = \frac{1}{4} - 5 = -\frac{19}{4} \approx -4.75
\]

2. **For \( x = 1 \)**:
\[
y = 2^{(1 - 2)} - 5 = 2^{-1} - 5 = \frac{1}{2} - 5 = -\frac{9}{2} = -4.5
\]

3. **For \( x = 2 \)**:
\[
y = 2^{(2 - 2)} - 5 = 2^{0} - 5 = 1 - 5 = -4
\]

4. **For \( x = 3 \)**:
\[
y = 2^{(3 - 2)} - 5 = 2^{1} - 5 = 2 - 5 = -3
\]

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

The integer coordinate points we can plot are:
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)

### Summary of Calculated Points
The integer coordinate points are:
1. \( (2, -4) \)
2. \( (3, -3) \)
3. \( (4, -1) \)

### Step 3: Graph the Function
Now, we will plot the points and include the asymptote at \( y = -5 \).

**Graph Overview:**
- The asymptote line: \( y = -5 \)
- Points to plot:
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)

**Graphing**:
You can graph this function using graphing software or a graphing calculator to visualize how the function approaches the asymptote \( y = -5 \) and includes the points listed above.

The graph would show the exponential growth on the right side, starting from the asymptote on the left and passing through the calculated points.

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