Questions: Write a linear function f with f(5)=7 and f(-2)=0.

Solution Steps

To find a linear function \( f(x) \) given two points, we need to determine the slope \( m \) and the y-intercept \( b \) of the line. The slope \( m \) can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Once we have the slope, we can use one of the points to solve for the y-intercept \( b \) using the equation \( y = mx + b \).

1. Calculate the slope \( m \) using the given points (5, 7) and (-2, 0).
2. Use the slope and one of the points to solve for the y-intercept \( b \).
3. Write the linear function \( f(x) \) in the form \( f(x) = mx + b \).

To find the slope \( m \) of the linear function, we use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points \( (5, 7) \) and \( (-2, 0) \):

\[ m = \frac{0 - 7}{-2 - 5} = \frac{-7}{-7} = 1.0 \]

Next, we calculate the y-intercept \( b \) using the slope and one of the points. We can use the point \( (5, 7) \):

\[ b = y_1 - m \cdot x_1 = 7 - 1.0 \cdot 5 = 7 - 5 = 2.0 \]

Now that we have both the slope and the y-intercept, we can write the linear function \( f(x) \):

\[ f(x) = mx + b = 1.0x + 2.0 \]

Final Answer