At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine if the table represents a linear function, we need to examine the relationship between the [tex]\( x \)[/tex]-values and the [tex]\( y \)[/tex]-values. Specifically, we should check if the differences between consecutive [tex]\( y \)[/tex]-values are consistent. If they are, the function is linear.
Here is the given table for reference:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 5 \\ \hline 2 & 10 \\ \hline 3 & 15 \\ \hline 4 & 20 \\ \hline 5 & 25 \\ \hline \end{array} \][/tex]
Now, let's calculate the differences between consecutive [tex]\( y \)[/tex]-values:
- The difference between [tex]\( y_2 \)[/tex] and [tex]\( y_1 \)[/tex] is [tex]\( 10 - 5 = 5 \)[/tex].
- The difference between [tex]\( y_3 \)[/tex] and [tex]\( y_2 \)[/tex] is [tex]\( 15 - 10 = 5 \)[/tex].
- The difference between [tex]\( y_4 \)[/tex] and [tex]\( y_3 \)[/tex] is [tex]\( 20 - 15 = 5 \)[/tex].
- The difference between [tex]\( y_5 \)[/tex] and [tex]\( y_4 \)[/tex] is [tex]\( 25 - 20 = 5 \)[/tex].
All these differences are [tex]\( 5 \)[/tex], which is consistent.
Since the differences between consecutive [tex]\( y \)[/tex]-values are the same (each equal to 5), we can conclude that the function described by this table is linear.
Moreover, a linear function can be expressed in the form [tex]\( y = mx + b \)[/tex]. To find the slope ([tex]\( m \)[/tex]), we can use any two of the points given in the table. Using the points [tex]\( (1, 5) \)[/tex] and [tex]\( (2, 10) \)[/tex]:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \][/tex]
Knowing the slope is [tex]\( 5 \)[/tex] and using one of the points [tex]\( (1, 5) \)[/tex], we can determine the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]). Plugging into the linear function formula [tex]\( y = mx + b \)[/tex]:
[tex]\[ 5 = 5 \cdot 1 + b \rightarrow 5 = 5 + b \rightarrow b = 0 \][/tex]
Therefore, the linear function is [tex]\( y = 5x \)[/tex].
Thus, the given table indeed represents a linear function.
Here is the given table for reference:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 5 \\ \hline 2 & 10 \\ \hline 3 & 15 \\ \hline 4 & 20 \\ \hline 5 & 25 \\ \hline \end{array} \][/tex]
Now, let's calculate the differences between consecutive [tex]\( y \)[/tex]-values:
- The difference between [tex]\( y_2 \)[/tex] and [tex]\( y_1 \)[/tex] is [tex]\( 10 - 5 = 5 \)[/tex].
- The difference between [tex]\( y_3 \)[/tex] and [tex]\( y_2 \)[/tex] is [tex]\( 15 - 10 = 5 \)[/tex].
- The difference between [tex]\( y_4 \)[/tex] and [tex]\( y_3 \)[/tex] is [tex]\( 20 - 15 = 5 \)[/tex].
- The difference between [tex]\( y_5 \)[/tex] and [tex]\( y_4 \)[/tex] is [tex]\( 25 - 20 = 5 \)[/tex].
All these differences are [tex]\( 5 \)[/tex], which is consistent.
Since the differences between consecutive [tex]\( y \)[/tex]-values are the same (each equal to 5), we can conclude that the function described by this table is linear.
Moreover, a linear function can be expressed in the form [tex]\( y = mx + b \)[/tex]. To find the slope ([tex]\( m \)[/tex]), we can use any two of the points given in the table. Using the points [tex]\( (1, 5) \)[/tex] and [tex]\( (2, 10) \)[/tex]:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \][/tex]
Knowing the slope is [tex]\( 5 \)[/tex] and using one of the points [tex]\( (1, 5) \)[/tex], we can determine the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]). Plugging into the linear function formula [tex]\( y = mx + b \)[/tex]:
[tex]\[ 5 = 5 \cdot 1 + b \rightarrow 5 = 5 + b \rightarrow b = 0 \][/tex]
Therefore, the linear function is [tex]\( y = 5x \)[/tex].
Thus, the given table indeed represents a linear function.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
