The problem provides a table of values for a function $f(x)$ and asks to estimate the value of $f(x)$ when $x=2$. The table contains the following points: $(-5, -7)$, $(3, -3)$, and $(5, -2)$.

1. Problem Description

The problem provides a table of values for a function $f(x)$ and asks to estimate the value of $f(x)$ when $x=2$. The table contains the following points: $(-5, -7)$, $(3, -3)$, and $(5, -2)$.

2. Solution Steps

Since we are given three points, we can estimate the function value at $x=2$ using linear interpolation between the two closest points. The value $x=2$ lies between $x=-5$ and $x=3$. However, since $x=2$ is closer to $x=3$, we will use the points $(3, -3)$ and $(-5, -7)$ to find the equation of the line and interpolate.

The slope $m$ of the line passing through points $(x_1, y_1) = (-5, -7)$ and $(x_2, y_2) = (3, -3)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-7)}{3 - (-5)} = \frac{-3 + 7}{3 + 5} = \frac{4}{8} = \frac{1}{2}$

Using the point-slope form of a line $y - y_1 = m(x - x_1)$, we can write the equation of the line using the point $(3, -3)$:

$y - (-3) = \frac{1}{2}(x - 3)$

$y + 3 = \frac{1}{2}x - \frac{3}{2}$

$y = \frac{1}{2}x - \frac{3}{2} - 3$

$y = \frac{1}{2}x - \frac{3}{2} - \frac{6}{2}$

$y = \frac{1}{2}x - \frac{9}{2}$

Now, we substitute $x=2$ into the equation to estimate $f(2)$:

$f(2) = \frac{1}{2}(2) - \frac{9}{2} = 1 - \frac{9}{2} = \frac{2}{2} - \frac{9}{2} = -\frac{7}{2} = -3.5$

3. Final Answer

$f(2) = -3.5$