Let's determine the slope and the [tex]\( y \)[/tex]-intercept of the linear function that fits the given table of coordinates.
The table of coordinates is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-\frac{3}{4} & -\frac{1}{30} \\
\hline
-\frac{1}{2} & -\frac{2}{15} \\
\hline
\frac{1}{4} & -\frac{13}{30} \\
\hline
\frac{2}{3} & -\frac{3}{5} \\
\hline
\end{array}
\][/tex]
Step 1: Calculate the slope (m)
The slope [tex]\( m \)[/tex] of a linear function can be found by averaging the slopes between consecutive points. To find the slope between any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on the line, we use the formula:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Step 2: Find slopes between consecutive points
Calculate the slopes between each pair of consecutive points:
1. For points [tex]\(\left( -\frac{3}{4}, -\frac{1}{30} \right)\)[/tex] and [tex]\(\left( -\frac{1}{2}, -\frac{2}{15} \right)\)[/tex]:
[tex]\[ m_1 = \frac{-\frac{2}{15} - \left( -\frac{1}{30} \right)}{-\frac{1}{2} - \left( -\frac{3}{4} \right)} \][/tex]
2. For points [tex]\(\left( -\frac{1}{2}, -\frac{2}{15} \right)\)[/tex] and [tex]\(\left( \frac{1}{4}, -\frac{13}{30} \right)\)[/tex]:
[tex]\[ m_2 = \frac{-\frac{13}{30} - \left( -\frac{2}{15} \right)}{\frac{1}{4} - \left( -\frac{1}{2} \right)} \][/tex]
3. For points [tex]\(\left( \frac{1}{4}, -\frac{13}{30} \right)\)[/tex] and [tex]\(\left( \frac{2}{3}, -\frac{3}{5} \right)\)[/tex]:
[tex]\[ m_3 = \frac{-\frac{3}{5} - \left( -\frac{13}{30} \right)}{\frac{2}{3} - \frac{1}{4}} \][/tex]
The average of these individual slopes will provide the overall slope [tex]\( m \)[/tex].
It turns out that with detailed calculations:
[tex]\[ m \approx -0.4 \][/tex]
Step 3: Calculate the y-intercept (b)
The equation of the line is [tex]\( y = mx + b \)[/tex]. We can use any point from the table to find [tex]\( b \)[/tex]. Using [tex]\(( -\frac{3}{4}, -\frac{1}{30} )\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ -\frac{1}{30} = -0.4 \left( -\frac{3}{4} \right) + b \][/tex]
[tex]\[ -\frac{1}{30} = 0.3 + b \][/tex]
[tex]\[ b = -\frac{1}{30} - 0.3 \][/tex]
[tex]\[ b \approx -0.333 \][/tex]
Hence, the slope [tex]\( m \)[/tex] is approximately [tex]\(-0.4\)[/tex] and the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\(-0.333\)[/tex].
Thus, the correct solution is:
The slope is [tex]\(-\frac{2}{5}\)[/tex] and the [tex]\( y \)[/tex]-intercept is [tex]\(-\frac{1}{3}\)[/tex].
