Questions: If f(x) is a linear function and given f(6)=7 and f(10)=-2, determine the linear function. a.) What is the slope? (Be sure to leave your answer in reduced fraction form.) b.) What is the y-intercept? (Be sure to leave your answer in reduced fraction form.) c.) What is f(x) ? f(x)=

Solution Steps

To determine the linear function $f(x)$ given two points, we need to:

1. Calculate the slope $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
2. Use the slope and one of the points to find the y-intercept $b$ using the formula $y = mx + b$.
3. Formulate the linear function $f(x) = mx + b$.

To find the slope $m$ of the linear function, we use the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Given the points $(6, 7)$ and $(10, -2)$: $$m = \frac{-2 - 7}{10 - 6} = \frac{-9}{4} = -2.25$$

Using the slope $m$ and one of the points, we can find the y-intercept $b$ using the formula: $$y = mx + b$$ Substituting $x = 6$, $y = 7$, and $m = -2.25$: $$7 = -2.25 \cdot 6 + b$$ $$7 = -13.5 + b$$ $$b = 7 + 13.5 = 20.5$$

With the slope $m = -2.25$ and the y-intercept $b = 20.5$, the linear function $f(x)$ is: $$f(x) = -2.25x + 20.5$$

Final Answer