Graph the line with slope -3/4 passing through the point (-5, -4).
Understand the Problem
The question asks to visually represent a straight line on a graph. We're given two pieces of information to achieve this: the slope of the line, which is -3/4, and a point that the line passes through, which is (-5, -4). The slope indicates how much the line rises (or falls if negative) for every unit of horizontal change, and knowing a point on the line fixes its position in the coordinate plane.
Answer
The line passes through $(-5, -4)$ with a slope of $\frac{3}{4}$.
Answer for screen readers
\begin{tikzpicture}
\begin{axis}[
axis lines = center,
xmin=-10, xmax=10,
ymin=-10, ymax=10,
xlabel=$x$, ylabel=$y$,
xtick={-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10},
ytick={-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10},
grid=both,
minor tick num=1,
width=8cm, height=8cm
]
\addplot[domain=-10:10, samples=100, thick] {3/4*x + 3/4*5 - 4};
\addplot[only marks, mark=*] coordinates {(-5,-4)};
\addplot[only marks, mark=*] coordinates {(-1,-1)};
\end{axis}
\end{tikzpicture}
Steps to Solve
- Plot the given point
Plot the point $(-5, -4)$ on the coordinate plane.
- Use the slope to find another point
The slope is $\frac{3}{4}$, which means for every 4 units you move to the right on the x-axis, you move 3 units up on the y-axis. Starting from the given point $(-5, -4)$, move 4 units to the right and 3 units up. $$(-5+4, -4+3) = (-1, -1)$$ So, the new point is $(-1, -1)$.
- Draw a line through the two points
Draw a straight line that passes through the points $(-5, -4)$ and $(-1, -1)$. This line represents the equation with the given slope and passing through the specified point.
\begin{tikzpicture}
\begin{axis}[
axis lines = center,
xmin=-10, xmax=10,
ymin=-10, ymax=10,
xlabel=$x$, ylabel=$y$,
xtick={-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10},
ytick={-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10},
grid=both,
minor tick num=1,
width=8cm, height=8cm
]
\addplot[domain=-10:10, samples=100, thick] {3/4*x + 3/4*5 - 4};
\addplot[only marks, mark=*] coordinates {(-5,-4)};
\addplot[only marks, mark=*] coordinates {(-1,-1)};
\end{axis}
\end{tikzpicture}
More Information
The equation of the line is $y = \frac{3}{4}x - \frac{1}{4}$. With a slope of $\frac{3}{4}$, the line rises 3 units for every 4 units moved to the right.
Tips
A common mistake is to misinterpret the slope. A positive slope means the line goes upwards from left to right. Make sure to move in the correct direction based on the sign of the slope. Another common mistake is incorrectly plotting the points or miscalculating the new point using the slope.
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