Question

The number of members, f(x), in a local swimming club increased by 30% every year over a period of x years. The function below shows the relationship between f(x) and x:

f(x) = 10(1.3)x

Which of the following graphs best represents the function?

graph of exponential function going up from left to right through the point approximately 0 comma 1 and approximately 6 tenths comma 5
graph of exponential function going up from left to right in quadrant 1 through the point 0, 0 and continuing toward infinity
graph of exponential function going up from left to right through the point 0 comma 10 and approximately 4 comma 30
graph of exponential function going up from left to right through the point 0 comma 1 and approximately 6 comma 5

Answer 1

The correct graph for the function f(x) = 10(1.3)^x, which models exponential growth, must start at the point (0, 10) and show a rise as x increases, without passing through the origin.

Step-by-step explanation:

The function given by f(x) = 10(1.3)x represents the number of members in a local swimming club that increases by 30% each year over a period of x years. Given the base information for this exponential function, when x = 0, the initial number of members is f(0) = 10. So the correct graph must pass through the point (0, 10).

Additionally, since it's an exponential growth function with a growth rate of 1.3, the graph will be rising as x increases, and will not pass through the origin (0, 0).

Instead, it will rise more steeply as x increases because the rate of increase is proportional to the current value.

Knowing this, the graph that best represents the function will be one in which the curve is moving upward from left to right and passes through the point (0, 10), indicating the initial value of the club's members. As x increases, the value of f(x) will grow at an increasing rate, demonstrating the exponential growth trend associated with the function.