Answer :
To determine which best describes the graph of the function [tex]\( f(x) = 4(1.5)^x \)[/tex], we need to analyze its key properties.
### Step-by-Step Analysis:
1. Identify the y-intercept:
- The y-intercept occurs when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 4(1.5)^0 \][/tex]
- Since any non-zero number raised to the power of 0 equals 1:
[tex]\[ f(0) = 4(1) = 4 \][/tex]
- Therefore, the graph passes through the point [tex]\( (0, 4) \)[/tex].
2. Determine the growth factor:
- The function [tex]\( f(x) = a \cdot b^x \)[/tex] is an exponential function where [tex]\( a \)[/tex] is the initial value (or y-intercept), and [tex]\( b \)[/tex] is the base or growth factor.
- In the function [tex]\( f(x) = 4(1.5)^x \)[/tex], the base [tex]\( b = 1.5 \)[/tex].
3. Interpret the growth factor:
- Each increase of 1 in the [tex]\( x \)[/tex]-values means the [tex]\( y \)[/tex]-values are multiplied by the growth factor [tex]\( b \)[/tex].
- For every increase of 1 in [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value is multiplied by [tex]\( 1.5 \)[/tex].
### Conclusion:
- The graph of the function [tex]\( f(x) = 4(1.5)^x \)[/tex] passes through the point [tex]\( (0, 4) \)[/tex].
- For each increase of 1 in the [tex]\( x \)[/tex]-values, the [tex]\( y \)[/tex]-values increase by a factor of [tex]\( 1.5 \)[/tex].
Thus, the correct description of the graph is:
- The graph passes through the point [tex]\((0,4)\)[/tex], and for each increase of 1 in the [tex]\( x \)[/tex]-values, the [tex]\( y \)[/tex]-values increase by a factor of 1.5.
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
### Step-by-Step Analysis:
1. Identify the y-intercept:
- The y-intercept occurs when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 4(1.5)^0 \][/tex]
- Since any non-zero number raised to the power of 0 equals 1:
[tex]\[ f(0) = 4(1) = 4 \][/tex]
- Therefore, the graph passes through the point [tex]\( (0, 4) \)[/tex].
2. Determine the growth factor:
- The function [tex]\( f(x) = a \cdot b^x \)[/tex] is an exponential function where [tex]\( a \)[/tex] is the initial value (or y-intercept), and [tex]\( b \)[/tex] is the base or growth factor.
- In the function [tex]\( f(x) = 4(1.5)^x \)[/tex], the base [tex]\( b = 1.5 \)[/tex].
3. Interpret the growth factor:
- Each increase of 1 in the [tex]\( x \)[/tex]-values means the [tex]\( y \)[/tex]-values are multiplied by the growth factor [tex]\( b \)[/tex].
- For every increase of 1 in [tex]\( x \)[/tex], the [tex]\( y \)[/tex]-value is multiplied by [tex]\( 1.5 \)[/tex].
### Conclusion:
- The graph of the function [tex]\( f(x) = 4(1.5)^x \)[/tex] passes through the point [tex]\( (0, 4) \)[/tex].
- For each increase of 1 in the [tex]\( x \)[/tex]-values, the [tex]\( y \)[/tex]-values increase by a factor of [tex]\( 1.5 \)[/tex].
Thus, the correct description of the graph is:
- The graph passes through the point [tex]\((0,4)\)[/tex], and for each increase of 1 in the [tex]\( x \)[/tex]-values, the [tex]\( y \)[/tex]-values increase by a factor of 1.5.
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
