To graph the function [tex]\( f(x) = -\frac{1}{4} x - 2 \)[/tex], we need to find at least two points on the line.
1. Choosing the first point:
- Select [tex]\( x = 0 \)[/tex], which is often a convenient choice because it's the y-intercept.
- Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = -\frac{1}{4}(0) - 2 = -2
\][/tex]
- This gives us the point [tex]\((0, -2)\)[/tex].
2. Choosing the second point:
- Select another value for [tex]\( x \)[/tex], say [tex]\( x = 4 \)[/tex].
- Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[
f(4) = -\frac{1}{4}(4) - 2 = -1 - 2 = -3
\][/tex]
- This gives us the point [tex]\((4, -3)\)[/tex].
Now we have our two points:
- The first point is [tex]\((0, -2)\)[/tex].
- The second point is [tex]\((4, -3)\)[/tex].
Using these points, we can graph the line [tex]\( f(x) = -\frac{1}{4} x - 2 \)[/tex]:
1. Plot the point [tex]\((0, -2)\)[/tex] on the coordinate plane.
2. Plot the point [tex]\((4, -3)\)[/tex] on the coordinate plane.
3. Draw a line through these two points, extending it in both directions. This line represents the graph of the function [tex]\( f(x) = -\frac{1}{4} x - 2 \)[/tex].
These two points determine the line, and you can see the slope and y-intercept directly from the formula. The graph shows a line that decreases as [tex]\( x \)[/tex] increases, consistent with the negative slope of [tex]\(-\frac{1}{4}\)[/tex].
