At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which table represents the graph of a logarithmic function with both an [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercept., let's translate the provided information into a mathematical context:
A logarithmic function of the form [tex]\( y = \log_b(x) \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex] and typically intersects the [tex]\( y \)[/tex]-axis when [tex]\( x = 1 \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex]. It generally intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = b^{-y} \)[/tex].
In the given table, we have the following data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0.5 & -0.631 \\ \hline 1.5 & 0.369 \\ \hline 2.5 & 0.834 \\ \hline 3.5 & 1.146 \\ \hline 4.5 & 1.369 \\ \hline \end{array} \][/tex]
From this table, we observe:
- For [tex]\( x = 0.5 \)[/tex], [tex]\( y = -0.631 \)[/tex]
- For [tex]\( x = 1.5 \)[/tex], [tex]\( y = 0.369 \)[/tex]
- For [tex]\( x = 2.5 \)[/tex], [tex]\( y = 0.834 \)[/tex]
- For [tex]\( x = 3.5 \)[/tex], [tex]\( y = 1.146 \)[/tex]
- For [tex]\( x = 4.5 \)[/tex], [tex]\( y = 1.369 \)[/tex]
Let's analyze these pairs:
1. The point [tex]\( (0.5, -0.631) \)[/tex] implies a negative [tex]\( y \)[/tex]-value, which is consistent with the logarithmic function, as [tex]\( \log_b(0.5) \)[/tex] is negative for any base [tex]\( b > 1 \)[/tex].
2. The point [tex]\( (1.5, 0.369) \)[/tex] suggests a positive [tex]\( y \)[/tex]-value, which indicates that for [tex]\( x > 1 \)[/tex], the function value is positive.
3. The value [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases, which is also characteristic of a logarithmic function when [tex]\( x > 1 \)[/tex].
Given these points, the table represents a logarithmic function, but we need to confirm if the intercepts coordinate with our bounds.
An intercept understanding:
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 1 \)[/tex], we typically intersect the [tex]\( y \)[/tex]-axis at [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex]-intercept: The table does not explicitly show [tex]\( x > 0 \)[/tex] and [tex]\( x \)[/tex] yielding exactly zero [tex]\( y \)[/tex].
From these, the given intercepts seem consistent with logarithmic characteristics.
Therefore, the table provided does represent the graph of a logarithmic function with both [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts.
A logarithmic function of the form [tex]\( y = \log_b(x) \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex] and typically intersects the [tex]\( y \)[/tex]-axis when [tex]\( x = 1 \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex]. It generally intersects the [tex]\( x \)[/tex]-axis at [tex]\( x = b^{-y} \)[/tex].
In the given table, we have the following data:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0.5 & -0.631 \\ \hline 1.5 & 0.369 \\ \hline 2.5 & 0.834 \\ \hline 3.5 & 1.146 \\ \hline 4.5 & 1.369 \\ \hline \end{array} \][/tex]
From this table, we observe:
- For [tex]\( x = 0.5 \)[/tex], [tex]\( y = -0.631 \)[/tex]
- For [tex]\( x = 1.5 \)[/tex], [tex]\( y = 0.369 \)[/tex]
- For [tex]\( x = 2.5 \)[/tex], [tex]\( y = 0.834 \)[/tex]
- For [tex]\( x = 3.5 \)[/tex], [tex]\( y = 1.146 \)[/tex]
- For [tex]\( x = 4.5 \)[/tex], [tex]\( y = 1.369 \)[/tex]
Let's analyze these pairs:
1. The point [tex]\( (0.5, -0.631) \)[/tex] implies a negative [tex]\( y \)[/tex]-value, which is consistent with the logarithmic function, as [tex]\( \log_b(0.5) \)[/tex] is negative for any base [tex]\( b > 1 \)[/tex].
2. The point [tex]\( (1.5, 0.369) \)[/tex] suggests a positive [tex]\( y \)[/tex]-value, which indicates that for [tex]\( x > 1 \)[/tex], the function value is positive.
3. The value [tex]\( y \)[/tex] is increasing as [tex]\( x \)[/tex] increases, which is also characteristic of a logarithmic function when [tex]\( x > 1 \)[/tex].
Given these points, the table represents a logarithmic function, but we need to confirm if the intercepts coordinate with our bounds.
An intercept understanding:
- [tex]\( y \)[/tex]-intercept: At [tex]\( x = 1 \)[/tex], we typically intersect the [tex]\( y \)[/tex]-axis at [tex]\( y = 0 \)[/tex].
- [tex]\( x \)[/tex]-intercept: The table does not explicitly show [tex]\( x > 0 \)[/tex] and [tex]\( x \)[/tex] yielding exactly zero [tex]\( y \)[/tex].
From these, the given intercepts seem consistent with logarithmic characteristics.
Therefore, the table provided does represent the graph of a logarithmic function with both [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
