To analyze and describe the graph of the function [tex]\( f(x) = 4 \cdot (1.5)^x \)[/tex], we can follow several key steps:
1. Identifying the Type of Function:
- This is an exponential function where the base is [tex]\( 1.5 \)[/tex] and the initial value (coefficient) is [tex]\( 4 \)[/tex].
2. General Properties of Exponential Functions:
- Exponential functions of the form [tex]\( a \cdot b^x \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex] will always increase as [tex]\( x \)[/tex] increases.
- This function will be continuously increasing because the base [tex]\( 1.5 \)[/tex] is greater than [tex]\( 1 \)[/tex].
3. Determining the Y-Intercept:
- The y-intercept is found by evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
- [tex]\( f(0) = 4 \cdot (1.5)^0 = 4 \cdot 1 = 4 \)[/tex].
- Hence, the function intersects the y-axis at the point [tex]\((0, 4)\)[/tex].
4. Behavior of the Function:
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] will grow exponentially, meaning it increases at an ever-accelerating rate.
- As [tex]\( x \)[/tex] decreases, [tex]\( f(x) \)[/tex] will approach zero but never actually reach zero, because the values of [tex]\( (1.5)^x \)[/tex] get smaller and smaller but remain positive.
Therefore, the graph of the function [tex]\( f(x) = 4 \cdot (1.5)^x \)[/tex] is an exponential curve that:
- Continuously increases.
- Has a y-intercept at [tex]\((0, 4)\)[/tex].
In conclusion, the function [tex]\( f(x) = 4 \cdot (1.5)^x \)[/tex] is best described by the statement:
"The graph of the function [tex]\( f(x) = 4(1.5)^x \)[/tex] is an exponential curve that continuously increases, with a y-intercept at (0, 4)."
