Question
Graph the exponential function y = 0.5 (6) **
. Identify the y-intercept. (1 point)
0.5
-4
1
. Identify the y-intercept. (1 point)
0.5
-4
1
Answers
Answered by
GPT-4o mini
To graph the exponential function \( y = 0.5 \cdot 6^x \) and identify the y-intercept, let’s follow these steps:
### Step 1: Understand the Function
The function \( y = 0.5 \cdot 6^x \) describes an exponential function where:
- The base is \( 6 \) (which is greater than 1), meaning the function will grow rapidly as \( x \) increases.
- The coefficient \( 0.5 \) indicates that the function starts at half of its base value at \( x = 0 \).
### Step 2: Find the Y-Intercept
To find the y-intercept, we need to evaluate the function at \( x = 0 \):
\[
y = 0.5 \cdot 6^0
\]
Since \( 6^0 = 1 \):
\[
y = 0.5 \cdot 1 = 0.5
\]
Thus, the y-intercept of the graph is \( (0, 0.5) \).
### Step 3: Sketch the Graph
1. **Plot the Y-Intercept**: Start by plotting the point \( (0, 0.5) \).
2. **Choose Additional Points**: You can calculate a few more points for better accuracy:
- For \( x = 1 \): \( y = 0.5 \cdot 6^1 = 0.5 \cdot 6 = 3 \) → Point \( (1, 3) \)
- For \( x = 2 \): \( y = 0.5 \cdot 6^2 = 0.5 \cdot 36 = 18 \) → Point \( (2, 18) \)
- For \( x = -1 \): \( y = 0.5 \cdot 6^{-1} = 0.5 \cdot \frac{1}{6} \approx 0.083 \) → Point \( (-1, 0.083) \)
3. **Draw the Curve**: Connect the points smoothly, showing that as \( x \) increases, \( y \) increases rapidly, and although \( y \) approaches \( 0 \) as \( x \) becomes more negative, it never reaches \( 0 \).
### Conclusion
The y-intercept of the function \( y = 0.5 \cdot 6^x \) is \( (0, 0.5) \). The graph will show an exponential growth starting from the y-intercept and rising steeply as \( x \) increases.
### Step 1: Understand the Function
The function \( y = 0.5 \cdot 6^x \) describes an exponential function where:
- The base is \( 6 \) (which is greater than 1), meaning the function will grow rapidly as \( x \) increases.
- The coefficient \( 0.5 \) indicates that the function starts at half of its base value at \( x = 0 \).
### Step 2: Find the Y-Intercept
To find the y-intercept, we need to evaluate the function at \( x = 0 \):
\[
y = 0.5 \cdot 6^0
\]
Since \( 6^0 = 1 \):
\[
y = 0.5 \cdot 1 = 0.5
\]
Thus, the y-intercept of the graph is \( (0, 0.5) \).
### Step 3: Sketch the Graph
1. **Plot the Y-Intercept**: Start by plotting the point \( (0, 0.5) \).
2. **Choose Additional Points**: You can calculate a few more points for better accuracy:
- For \( x = 1 \): \( y = 0.5 \cdot 6^1 = 0.5 \cdot 6 = 3 \) → Point \( (1, 3) \)
- For \( x = 2 \): \( y = 0.5 \cdot 6^2 = 0.5 \cdot 36 = 18 \) → Point \( (2, 18) \)
- For \( x = -1 \): \( y = 0.5 \cdot 6^{-1} = 0.5 \cdot \frac{1}{6} \approx 0.083 \) → Point \( (-1, 0.083) \)
3. **Draw the Curve**: Connect the points smoothly, showing that as \( x \) increases, \( y \) increases rapidly, and although \( y \) approaches \( 0 \) as \( x \) becomes more negative, it never reaches \( 0 \).
### Conclusion
The y-intercept of the function \( y = 0.5 \cdot 6^x \) is \( (0, 0.5) \). The graph will show an exponential growth starting from the y-intercept and rising steeply as \( x \) increases.