Sure! Let's graph the function [tex]\( h(x)=6 \cdot\left(\frac{4}{3}\right)^x \)[/tex] step-by-step.
### Step 1: Understanding the function
The given function is an exponential function. The base of the exponential term is greater than 1, which means the function will show exponential growth.
- The base of the exponent is [tex]\( \frac{4}{3} \)[/tex].
- The coefficient 6 controls the initial value when [tex]\( x=0 \)[/tex].
### Step 2: Creating a table of values
To graph the function, we can start by calculating some values of [tex]\( h(x) \)[/tex] for selected [tex]\( x \)[/tex] values.
| [tex]\( x \)[/tex] | [tex]\( h(x) = 6 \cdot \left( \frac{4}{3}\right)^x \)[/tex] |
|------|-------------------------------------------|
| -3 | [tex]\( 6 \cdot \left( \frac{4}{3} \right)^{-3} = 6 \cdot \left( \frac{3}{4} \right)^3 = 6 \cdot \frac{27}{64} \approx 2.53 \)[/tex] |
| -2 | [tex]\( 6 \cdot \left( \frac{4}{3} \right)^{-2} = 6 \cdot \left( \frac{3}{4} \right)^2 = 6 \cdot \frac{9}{16} = 6 \cdot 0.5625 = 3.375 \)[/tex] |
| -1 | [tex]\( 6 \cdot \left( \frac{4}{3} \right)^{-1} = 6 \cdot \left( \frac{3}{4} \right) = 6 \cdot 0.75 = 4.5 \)[/tex] |
| 0 | [tex]\( 6 \cdot 1 = 6 \)[/tex] |
| 1 | [tex]\( 6 \cdot \left( \frac{4}{3} \right) = 6 \cdot 1.333 = 8 \)[/tex] |
| 2 | [tex]\( 6 \cdot \left( \frac{4}{3} \right)^2 = 6 \cdot 1.777 = 10.67 \)[/tex] |
| 3 | [tex]\( 6 \cdot \left( \frac{4}{3} \right)^3 = 6 \cdot 2.37 = 14.22 \)[/tex] |
### Step 3: Plotting the points
We will now plot these points on a Cartesian plane:
- At [tex]\( x = -3 \)[/tex], [tex]\( h(x) \approx 2.53 \)[/tex]
- At [tex]\( x = -2 \)[/tex], [tex]\( h(x) = 3.375 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( h(x) = 4.5 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 6 \)[/tex]
- At [tex]\( x = 1 \)[/tex], [tex]\( h(x) = 8 \)[/tex]
- At [tex]\( x = 2 \)[/tex], [tex]\( h(x) = 10.67 \)[/tex]
- At [tex]\( x = 3 \)[/tex], [tex]\( h(x) = 14.22 \)[/tex]
### Step 4: Drawing the graph
1. Draw the horizontal axis (x-axis) and the vertical axis (y-axis).
2. Label the axes with appropriate scales. Since the y-values grow exponentially, we may need a larger range for the y-axis.
3. Plot the points calculated above on the graph.
4. Finally, draw a smooth curve through the points to represent the exponential function.
### Visual Representation
Here is how you might envision the plot:
- The graph will pass through the points mentioned, showing an exponential rise as [tex]\( x \)[/tex] increases.
- For [tex]\( x < 0 \)[/tex], the function values will get closer and closer to zero but remain positive (since the exponential function never touches the x-axis).
### Additional Points
- The y-intercept is at (0, 6) since [tex]\( h(0) = 6 \)[/tex].
- The graph will asymptote towards the x-axis but never touch or cross it.
By joining these points with a smooth curve, you will obtain the graph of the function [tex]\( h(x) = 6 \cdot \left( \frac{4}{3} \right)^x \)[/tex]. As [tex]\( x \)[/tex] becomes more positive, [tex]\( h(x) \)[/tex] increases rapidly, illustrating the nature of exponential growth.
