Sure, let's go through the steps to graph the line [tex]\( y = -\frac{1}{6} x + 1 \)[/tex].
### Step-by-Step Solution
1. Identify the slope and y-intercept:
- The equation of the line is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In the equation [tex]\( y = -\frac{1}{6} x + 1 \)[/tex]:
- The slope ([tex]\( m \)[/tex]) is [tex]\( -\frac{1}{6} \)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( 1 \)[/tex].
2. Plot the y-intercept:
- This is the point where the line crosses the y-axis. For our equation, the y-intercept is [tex]\( 1 \)[/tex], so plot the point (0, 1) on the graph.
3. Use the slope to find another point:
- The slope [tex]\( -\frac{1}{6} \)[/tex] means that for every 6 units you move to the right along the x-axis (positive direction), the line moves down 1 unit (since the slope is negative).
- Starting from the y-intercept (0, 1), move 6 units to the right (along the x-axis) to [tex]\( x = 6 \)[/tex]. From that point, move 1 unit down to [tex]\( y = 1 - 1 = 0 \)[/tex].
- This gives you the point (6, 0).
4. Draw the line:
- With the points (0, 1) and (6, 0) plotted, you can draw a straight line through these points. Extend the line in both directions to cover the entire graph.
### Example Plot Illustration
Here's how the points and line would look on a graph:
1. Plotting the y-intercept (0, 1):
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 1 \\
\end{array}
\][/tex]
2. Finding another point using the slope, from [tex]\( x = 0 \)[/tex] to [tex]\( x = 6 \)[/tex]:
Starting from (0, 1):
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
6 & 0 \\
\end{array}
\][/tex]
3. Drawing the line:
[tex]\[
\begin{tikzpicture}
\begin{axis}[
axis lines = middle,
xlabel = $x$,
ylabel = $y$,
]
% Plot the line
\addplot [
domain=-10:10,
samples=100,
color=blue,
]
{-1/6 * x + 1};
\end{axis}
\end{tikzpicture}
\][/tex]
By following these steps, you'll have successfully graphed the line for the equation [tex]\( y = -\frac{1}{6} x + 1 \)[/tex].
