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 that fit on the axes below.

Answer 1

To analyze and graph the function \( y = 2^{(x - 2)} - 5 \), let's begin with the following steps:

### Step 1: Determine the Asymptote
The function \( y = 2^{(x - 2)} - 5 \) is an exponential function that is shifted down by 5 units. The horizontal asymptote can be determined by considering the limit of the function as \( x \) approaches negative infinity:

\[
\lim_{x \to -\infty} (2^{(x - 2)} - 5) = 0 - 5 = -5
\]

So, the equation of the asymptote is:
\[
y = -5
\]

### Step 2: Find Points on the Graph
To plot the function, we can substitute various integer values of \( x \) to find corresponding \( y \) values. Below are calculations for several integer values of \( x \):

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

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

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

### Summary of Points
Based on these calculations, we have the following integer coordinate points:

- \( (0, -4.75) \)
- \( (1, -4.5) \)
- \( (2, -4) \)
- \( (3, -3) \)
- \( (4, -1) \)

### Step 3: Create a Graph
To graph this function, plot the points on the Cartesian coordinate system and draw the horizontal asymptote \( y = -5 \).

### Points Summary for Graphing
Here are the points you can plot on a coordinate grid, rounding the y-values where necessary:

1. \( (0, -5) \) (Approx)
2. \( (1, -5) \) (Approx)
3. \( (2, -4) \)
4. \( (3, -3) \)
5. \( (4, -1) \)

You can plot these points, and connect them smoothly to illustrate the exponential shape, approaching the asymptote \( y = -5 \) as \( x \) decreases.

Your final output for the graph would include these points plotted on the axes below, with a dashed line for the asymptote at \( y = -5 \).