To find the equation of the linear function represented by the table, we can use the two points provided: (18, 10) and (6, -3).
- Calculate the slope (m):
The formula for slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (18, 10) \) and \( (6, -3) \):
\[ m = \frac{-3 - 10}{6 - 18} = \frac{-13}{-12} = \frac{13}{12} \]
- Use the point-slope form of the equation:
The point-slope form is:
\[ y - y_1 = m(x - x_1) \]
We can use either point; let’s use \( (18, 10) \):
\[ y - 10 = \frac{13}{12}(x - 18) \]
- Rearranging to slope-intercept form (y = mx + b):
Expanding the equation:
\[ y - 10 = \frac{13}{12}x - \frac{13}{12} \times 18 \]
Calculating \( \frac{13}{12} \times 18 \):
\[ \frac{13 \times 18}{12} = \frac{234}{12} = 19.5 \]
So:
\[ y - 10 = \frac{13}{12}x - 19.5 \]
Now, add 10 to both sides:
\[ y = \frac{13}{12}x - 19.5 + 10 \]
Since 19.5 equals 19.5 and 10 combined gives \(-9.5\):
Converting -9.5 into a fraction gives \(-\frac{19}{2}\).
Thus:
\[ y = \frac{13}{12}x - \frac{19}{2} \]
- Matching with the provided responses:
The correct equation is:
\[ y = \left(\frac{13}{12}\right)x - \left(\frac{19}{2}\right) \]
Therefore, the answer is:
y = (1312)x − (192)