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Sagot :
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].
### 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].
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