Which best describes the graph of the function $f(x)=4(1.5)^x$?
The graph passes through the point $(0,4)$, and for each increase of 1 in the $x$-values, the $y$-values increase by 1.5.
The graph passes through the point $(0,4)$, and for each increase of 1 in the $x$-values, the $y$-values increase by a factor of 1.5.
The graph passes through the point $(0,1.5)$, and for each increase of 1 in the $x$-values, the $y$-values increase by 4.
The graph passes through the point $(0,1.5)$, and for each increase of 1 in the $x$-values, the $y$-values increase by a factor of 4.
Answer (1)
The graph passes through ( 0 , 4 ) since f ( 0 ) = 4 ( 1.5 ) 0 = 4 .
When x increases by 1, f ( x ) is multiplied by 1.5, since f ( x + 1 ) = 4 ( 1.5 ) x + 1 = 1.5".4 ( 1.5 ) x = 1.5 f ( x ) .
Therefore, for each increase of 1 in the x -values, the y -values increase by a factor of 1.5.
The correct description is: The graph passes through the point ( 0 , 4 ) , and for each increase of 1 in the x -values, the y -values increase by a factor of 1.5. T h e g r a p h p a sses t h ro ug h t h e p o in t ( 0 , 4 ) , an df ore a c hin cre a seo f 1 in t h e x − v a l u es , t h ey − v a l u es in cre a se b y a f a c t oro f 1.5.
Explanation
Understanding the Function We are given the function f ( x ) = 4 ( 1.5 ) x and asked to describe its graph. We need to determine the y-intercept and how the y-values change as x increases by 1.
Finding the y-intercept To find the y-intercept, we evaluate f ( 0 ) : f ( 0 ) = 4 ( 1.5 ) 0 = 4 ( 1 ) = 4 So, the graph passes through the point ( 0 , 4 ) .
Determining the Factor To determine how the y-values change as x increases by 1, we can evaluate f ( x + 1 ) and compare it to f ( x ) . f ( x + 1 ) = 4 ( 1.5 ) x + 1 = 4 ( 1.5 ) x ⋅ 1.5 = f ( x ) ".1.5 This shows that for each increase of 1 in the x-values, the y-values are multiplied by a factor of 1.5.
Conclusion Therefore, the graph passes through the point ( 0 , 4 ) , and for each increase of 1 in the x -values, the y -values increase by a factor of 1.5.
Examples
Exponential functions like f ( x ) = 4 ( 1.5 ) x are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For instance, if you invest $4000 in an account that earns 50% interest annually, the amount of money you have after x years can be modeled by this function (with the initial investment in thousands of dollars). Understanding the behavior of exponential functions helps in making predictions and informed decisions in finance, biology, and other fields.
