To graph the equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex], follow these steps:
1. Identify the slope and y-intercept:
- The equation 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.
- Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\( 1 \)[/tex].
2. Plot the y-intercept:
- The y-intercept is the point where the line crosses the y-axis. For [tex]\( y = -\frac{2}{3}x + 1 \)[/tex], the intercept is [tex]\( (0, 1) \)[/tex].
- Plot the point [tex]\( (0, 1) \)[/tex] on the graph.
3. Use the slope to find another point:
- The slope [tex]\( -\frac{2}{3} \)[/tex] means that for every 3 units you move horizontally to the right, you move 2 units down vertically.
- Starting from the y-intercept [tex]\( (0, 1) \)[/tex]:
- Move 3 units to the right: [tex]\( 0 + 3 = 3 \)[/tex].
- Move 2 units down: [tex]\( 1 - 2 = -1 \)[/tex].
- This gives the point [tex]\( (3, -1) \)[/tex].
- Plot the point [tex]\( (3, -1) \)[/tex].
4. Draw the line:
- Use a ruler to draw a straight line through the points [tex]\( (0, 1) \)[/tex] and [tex]\( (3, -1) \)[/tex].
- Extend the line in both directions and add arrows to indicate that it continues indefinitely.
5. Label the line:
- Optionally, label the line with its equation [tex]\( y = -\frac{2}{3}x + 1 \)[/tex].
Here is a summary:
1. Plot the y-intercept at [tex]\( (0, 1) \)[/tex].
2. Use the slope [tex]\( -\frac{2}{3} \)[/tex] to determine another point, [tex]\( (3, -1) \)[/tex].
3. Draw a line through these points, extending it in both directions.
An accurate graph would look like this:
[tex]\[
\begin{array}{l}
\begin{tikzpicture}
\begin{axis}[
axis lines = middle,
xlabel = $x$,
ylabel = $y$,
grid = both,
grid style = {gray!30},
width=8cm,
height=6cm,
ymin=-4, ymax=4,
xmin=-4, xmax=4,
]
\addplot [domain = -4:4, samples = 2, very thick] {-(2/3)*x + 1};
\node[pin=135:{$(0, 1)$},circle,fill,inner sep=1pt] at (axis cs: 0,1) {};
\node[pin=-45:{$(3, -1)$},circle,fill,inner sep=1pt] at (axis cs: 3,-1) {};
\end{axis}
\end{tikzpicture}
\end{array}
\][/tex]
By following these steps, you can graph the line that represents [tex]\( y = -\frac{2}{3}x + 1 \)[/tex] accurately.
