An art collector purchased a painting at an auction. The function shown represents the value of the painting in dollars, y, after x years.
Y= 575(1.045)^x
Which statement about the graph of the function are true?
Select TWO correct answers.
A. The x-intercept of the graph is 575.
B. The graph has an asymptote at x=0.
C. The y-intercept of the graph is 575.
D. The graph decreases as the value of x increases.
E. The graph has an asymptote at y=0.
The correct answers are A and E.
A. The x-intercept of the graph is when y = 0. Plugging in y = 0 into the equation, we get 0 = 575(1.045)^x. Since no non-zero number can be raised to the power of x and equal 0, there is no x-intercept. Therefore, statement A is false.
B. The graph has an asymptote at x = 0. As x approaches negative infinity, the value of y approaches positive infinity. Therefore, there is no asymptote at x = 0. So, statement B is false.
C. The y-intercept of the graph is when x = 0. Plugging in x = 0 into the equation, we get y = 575(1.045)^0 = 575. Therefore, the y-intercept is 575. So, statement C is true.
D. The graph decreases as the value of x increases. Since the base of the exponential function, 1.045, is greater than 1, raising it to a larger exponent will result in a larger number. Therefore, the graph increases as the value of x increases. So, statement D is false.
E. The graph has an asymptote at y = 0. As x approaches positive infinity, the value of y approaches positive infinity. Therefore, the graph does not have an asymptote at y = 0. So, statement E is false.