Which graph represents the function $y=5 \cdot\left(\frac{1}{3}\right)^x $?
Answer (1)
The function is an exponential function with a base between 0 and 1, so it is decreasing.
The y-intercept is 5.
The x-axis is a horizontal asymptote.
The graph represents a decreasing exponential function with a y-intercept of 5 and a horizontal asymptote at y=0.
Explanation
Analyze the function We are given the function y = 5" , ( 3 1 ) x and asked to identify its graph. Let's analyze the properties of this function to help us choose the correct graph.
Determine if the function is increasing or decreasing First, notice that this is an exponential function of the form y = a " , b x , where a = 5 and b = 3 1 . Since 0 < b < 1 , the function is decreasing. This means that as x increases, y decreases.
Find the y-intercept Next, let's find the y-intercept. The y-intercept is the value of y when x = 0 . So, we have y = 5" , ( 3 1 ) 0 = 5" , 1 = 5 . Thus, the y-intercept is 5.
Find the horizontal asymptote Now, let's consider the behavior of the function as x approaches infinity. As x → ∞ , ( 3 1 ) x → 0 , so y = 5" , ( 3 1 ) x → 0 . This means that the x-axis ( y = 0 ) is a horizontal asymptote.
Consider the behavior as x approaches negative infinity Finally, let's consider the behavior of the function as x approaches negative infinity. As x → − ∞ , ( 3 1 ) x → ∞ , so y = 5" , ( 3 1 ) x → ∞ .
Summarize the properties of the function In summary, the function is decreasing, has a y-intercept of 5, and has a horizontal asymptote at y = 0 . The graph should start at large y values for negative x, decrease as x increases, and approach the x-axis as x goes to infinity.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that earns compound interest, the amount of money you have over time can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled by an exponential function. Understanding the properties of exponential functions allows us to make predictions about these phenomena.
