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