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