which table represents the graph of a logarithmic function with both an x - and y - intercept?\ndone
which table represents the graph of a logarithmic function with both an x - and y - intercept?\ndone
Answer
Explanation:
Step1: Recall the domain of logarithmic functions
The domain of a logarithmic function (y = \log_b(x - h)+k) is (x>h). Logarithmic functions are not defined for non - positive values of the argument of the logarithm (when (b>0,b\neq1)). So, we can eliminate the table with (x=-1.5) and (x = - 0.5) (the second table) since the argument of a basic logarithmic function (y=\log(x)) has a domain (x>0) and even for a transformed function (y=\log_b(x - h)+k), the values of (x) for which (x - h\leq0) are not in the domain.
Step2: Recall the (x) and (y) - intercepts
The (x) - intercept of a function (y = f(x)) is a value of (x) such that (y = 0). The (y) - intercept is a value of (y) when (x = 0). For the first table:
- The (x) - intercept is (x = 3) (since (y = 0) when (x = 3)). But if we assume a general form (y=\log_b(x - h)+k), when (x = 4,y=-15) is an extreme value. Logarithmic functions (y=\log_b(x)) have a relatively slow rate of change. For the third table:
- We can check the general behavior of a logarithmic function (y=\log(x)) which has a vertical asymptote at (x = 0). A transformed logarithmic function (y=\log(x - h)+k). The third table has values of (x>0) (satisfying the domain condition for a basic - type logarithmic function (y=\log(x)) or (y = a\log(x)+b)).
- We can also check the rate of change. The difference in (y) - values:
- When (x) changes from (x_1=0.5) to (x_2 = 1.5) ((\Delta x=1)), (y) changes from (y_1=-0.631) to (y_2 = 0.369) ((\Delta y=0.369-(-0.631)=1)). When (x) changes from (x_2 = 1.5) to (x_3=2.5) ((\Delta x = 1)), (y) changes from (y_2=0.369) to (y_3 = 0.834) ((\Delta y=0.834 - 0.369=0.465)). When (x) changes from (x_3=2.5) to (x_4=3.5) ((\Delta x = 1)), (y) changes from (y_3=0.834) to (y_4 = 1.146) ((\Delta y=1.146 - 0.834 = 0.312)). This is consistent with the slow - growth rate of a logarithmic function (y=\log(x)) (after vertical and horizontal transformations).
Answer:
The third table.