Question
Graph the exponential function y equals 0.5(6)^x identified y intercept
Answers
Answered by
GPT-4o mini
To graph the exponential function \( y = 0.5(6)^x \) and identify its y-intercept, follow these steps:
1. **Identify the y-intercept:**
The y-intercept occurs when \( x = 0 \).
Substitute \( x = 0 \) into the function:
\[
y = 0.5(6)^0 = 0.5(1) = 0.5
\]
So the y-intercept is at the point \( (0, 0.5) \).
2. **Plotting Points:**
It can be helpful to calculate a few more points to get an accurate graph. Here are some values for \( x \):
- When \( x = -1 \):
\[
y = 0.5(6)^{-1} = 0.5 \cdot \frac{1}{6} = \frac{0.5}{6} \approx 0.0833
\]
- When \( x = 1 \):
\[
y = 0.5(6)^{1} = 0.5 \cdot 6 = 3
\]
- When \( x = 2 \):
\[
y = 0.5(6)^{2} = 0.5 \cdot 36 = 18
\]
So the points to plot are:
- \( (-1, 0.0833) \)
- \( (0, 0.5) \)
- \( (1, 3) \)
- \( (2, 18) \)
3. **Sketching the Graph:**
- Start by plotting the points calculated above.
- The graph of the function will be an exponential curve that rises steeply as \( x \) increases.
- The curve approaches the x-axis (but never touches it) as \( x \) decreases towards negative infinity (it approaches \( y = 0 \)).
4. **Graph Characteristics:**
- The function is always positive, \( y > 0 \) for all \( x \).
- It has a horizontal asymptote at \( y = 0 \).
- As \( x \) goes to infinity, \( y \) increases extremely rapidly.
Here is a representation of the graph based on the points plotted:
```
y
|
| * (2, 18)
| *
| *
| * (1, 3)
| *
|* (0, 0.5)
|__________________________ x
-1 0 1 2
```
In a proper graphing software or graph paper, you would see a smooth curve that rises steeply to the right and approaches the x-axis to the left. The y-intercept \( (0, 0.5) \) is clearly marked.
1. **Identify the y-intercept:**
The y-intercept occurs when \( x = 0 \).
Substitute \( x = 0 \) into the function:
\[
y = 0.5(6)^0 = 0.5(1) = 0.5
\]
So the y-intercept is at the point \( (0, 0.5) \).
2. **Plotting Points:**
It can be helpful to calculate a few more points to get an accurate graph. Here are some values for \( x \):
- When \( x = -1 \):
\[
y = 0.5(6)^{-1} = 0.5 \cdot \frac{1}{6} = \frac{0.5}{6} \approx 0.0833
\]
- When \( x = 1 \):
\[
y = 0.5(6)^{1} = 0.5 \cdot 6 = 3
\]
- When \( x = 2 \):
\[
y = 0.5(6)^{2} = 0.5 \cdot 36 = 18
\]
So the points to plot are:
- \( (-1, 0.0833) \)
- \( (0, 0.5) \)
- \( (1, 3) \)
- \( (2, 18) \)
3. **Sketching the Graph:**
- Start by plotting the points calculated above.
- The graph of the function will be an exponential curve that rises steeply as \( x \) increases.
- The curve approaches the x-axis (but never touches it) as \( x \) decreases towards negative infinity (it approaches \( y = 0 \)).
4. **Graph Characteristics:**
- The function is always positive, \( y > 0 \) for all \( x \).
- It has a horizontal asymptote at \( y = 0 \).
- As \( x \) goes to infinity, \( y \) increases extremely rapidly.
Here is a representation of the graph based on the points plotted:
```
y
|
| * (2, 18)
| *
| *
| * (1, 3)
| *
|* (0, 0.5)
|__________________________ x
-1 0 1 2
```
In a proper graphing software or graph paper, you would see a smooth curve that rises steeply to the right and approaches the x-axis to the left. The y-intercept \( (0, 0.5) \) is clearly marked.