Which table represents the graph of a logarithmic function with both an [tex]$x$[/tex]- and [tex]$y$[/tex]-intercept?
Answer (2)
Table 1 has an x-intercept but no y-intercept.
Table 2 does not represent a logarithmic function because it includes negative x values.
Table 3 has both an x-intercept and a y-intercept.
Therefore, the table that represents the graph of a logarithmic function with both an x -and y -intercept is Table 3, so the answer is T ab l e 3 .
Explanation
Understanding the Problem We are given three tables of x and y values and we want to determine which table represents a logarithmic function with both an x -intercept and a y -intercept. A logarithmic function has the general form y = a lo g b ( x − c ) + d , where a , b , c , and d are constants and 0"> b > 0 , b = 1 . A logarithmic function has a vertical asymptote at x = c . An x -intercept occurs when y = 0 , and a y -intercept occurs when x = 0 .
Analyzing the Tables Let's analyze each table to see if it could represent a logarithmic function. For a table to represent a logarithmic function, the x values must be greater than some constant c due to the vertical asymptote. We need to check if the table has both an x -intercept (a point where y = 0 ) and a y -intercept (a point where x = 0 ).
Analyzing Table 1 Table 1: The first x value is 3, and the corresponding y value is undefined. This suggests a vertical asymptote at or near x = 3 . The table has a y value of 0.585 when x = 5 , 1.322 when x = 6 , and 1.807 when x = 7 . There is no y -intercept because x starts at 3. There is an x -intercept between x = 4 and x = 5 since the y value changes sign (from -15 to 0.585). Thus, this table has an x -intercept but no y -intercept.
Analyzing Table 2 Table 2: The x values are -1.5, -0.5, 0.5, 1.5, 2.5. There is a y -intercept at x = 0 if we extrapolate. However, logarithmic functions are not defined for negative x values. There is an x -intercept between x = − 0.5 and x = 0.5 since the y value changes sign (from -1 to 0.585). However, there is no x -intercept since none of the y values are zero. Also, since the domain of a logarithmic function is c"> x > c , this table cannot represent a logarithmic function.
Analyzing Table 3 Table 3: The x values are 0.5, 1.5, 2.5, 3.5, 4.5. There is a y -intercept if we extrapolate to x = 0 . There is an x -intercept between x = 0.5 and x = 1.5 since the y value changes sign (from -0.631 to 0.369). Since the y value changes sign between x = 0.5 and x = 1.5 , there must be an x -intercept in this interval. Also, since the y value changes sign between x = 0.5 and x = 0 , there must be a y -intercept in this interval. Thus, this table has both an x -intercept and a y -intercept.
Conclusion Therefore, the third table is the only one that could represent a logarithmic function with both an x -intercept and a y -intercept.
Examples
Logarithmic functions are used to model many real-world phenomena, such as the loudness of sound (decibels), the intensity of earthquakes (Richter scale), and the growth of populations. Understanding the properties of logarithmic functions, such as intercepts and asymptotes, is crucial for interpreting and analyzing these models. For example, if we have data on the population of a city over time, we can use a logarithmic function to model the growth and predict when the population will reach a certain level. The x and y intercepts can tell us important information about the initial population and the time it takes to reach certain milestones.
Table 3 is the only table that represents a logarithmic function with both an x-intercept and a y-intercept. This is because it has no negative x values, indicating it is consistent with the properties of logarithmic functions. In addition, it has a sign change in y values, showcasing both intercepts.
;
