Questions: If f(x) is a linear function, f(-4)=-4, and f(5)=-2, find an equation for f(x). f(x)=

Solution Steps

To find the equation of a linear function given two points, we can use the point-slope form of a line. First, calculate the slope using the two given points. Then, use one of the points and the slope to write the equation in point-slope form, and finally convert it to slope-intercept form.

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

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

Substituting the given points \( (-4, -4) \) and \( (5, -2) \):

\[ m = \frac{-2 - (-4)}{5 - (-4)} = \frac{2}{9} \approx 0.2222 \]

Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with point \( (-4, -4) \):

\[ y - (-4) = \frac{2}{9}(x - (-4)) \]

This simplifies to:

\[ y + 4 = \frac{2}{9}(x + 4) \]

To convert to slope-intercept form \( y = mx + b \), we first distribute:

\[ y + 4 = \frac{2}{9}x + \frac{8}{9} \]

Now, isolate \( y \):

\[ y = \frac{2}{9}x + \frac{8}{9} - 4 \]

Converting \( -4 \) to a fraction with a common denominator:

\[ -4 = -\frac{36}{9} \]

Thus, we have:

\[ y = \frac{2}{9}x + \frac{8}{9} - \frac{36}{9} = \frac{2}{9}x - \frac{28}{9} \]

Final Answer