\( P(-2,1) \) and \( Q(6,8) \) be two points in the coordinate plane. a) Plot \( P \) and \( Q \) in the coordinate plane.

Certainly! Let's plot the points \( P(-2, 1) \) and \( Q(6, 8) \) on the coordinate plane. ### Step-by-Step Guide to Plotting Points 1. **Draw the Coordinate Axes:** - Draw a horizontal line for the **x-axis** and a vertical line for the **y-axis**. Label the intersection as the **origin (0, 0)**. - Decide on an appropriate scale for both axes (e.g., each unit represents 1 unit in the plane). 2. **Plot Point \( P(-2, 1) \):** - **X-coordinate (-2):** Since it's negative, move 2 units to the **left** of the origin along the x-axis. - **Y-coordinate (1):** From that position, move 1 unit **up** along the y-axis. - **Mark the Point:** Place a dot at this location and label it as \( P(-2, 1) \). 3. **Plot Point \( Q(6, 8) \):** - **X-coordinate (6):** Move 6 units to the **right** of the origin along the x-axis. - **Y-coordinate (8):** From that position, move 8 units **up** along the y-axis. - **Mark the Point:** Place a dot at this location and label it as \( Q(6, 8) \). ### Visual Representation Below is a simple representation of the coordinate plane with points \( P \) and \( Q \) plotted. ``` y | 10 | 9 | 8 | Q (6,8) 7 | 6 | 5 | 4 | 3 | 2 | 1 | P (-2,1) 0 |---------------------------- | -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 x ``` ### Using LaTeX with TikZ for a Precise Plot If you prefer a more precise and visual plot using LaTeX, you can use the TikZ package. Here's a sample code that you can compile in a LaTeX editor to generate the coordinate plane with the points plotted: ```latex \documentclass{standalone} \usepackage{tikz} \begin{document} \begin{tikzpicture}[scale=0.5] % Draw axes \draw[->] (-5,0) -- (8,0) node[right] {$x$}; \draw[->] (0,-1) -- (0,10) node[above] {$y$}; % Draw grid \draw[step=1cm,gray,very thin] (-5,-1) grid (8,10); % Plot point P(-2,1) \filldraw[red] (-2,1) circle (5pt) node[below left] {$P(-2,1)$}; % Plot point Q(6,8) \filldraw[blue] (6,8) circle (5pt) node[above right] {$Q(6,8)$}; % Optional: dashed lines to axes \draw[dashed] (-2,0) -- (-2,1); \draw[dashed] (0,1) -- (-2,1); \draw[dashed] (6,0) -- (6,8); \draw[dashed] (0,8) -- (6,8); \end{tikzpicture} \end{document} ``` **Explanation of the Code:** - **Axes:** The `\draw[->]` commands draw the x and y axes with arrows. - **Grid:** The `\draw[step=1cm,gray,very thin]` command creates a light grid to help visualize the points' locations. - **Points:** - Point \( P(-2, 1) \) is plotted in **red** and labeled accordingly. - Point \( Q(6, 8) \) is plotted in **blue** and labeled accordingly. - **Dashed Lines:** These help in visualizing the projection of each point onto the axes. ### Final Plot After compiling the above LaTeX code, you'll obtain a clear coordinate plane with both points \( P \) and \( Q \) accurately plotted and labeled. --- Feel free to use the method that best suits your needs! If you have access to graphing tools like Desmos, GeoGebra, or even graph paper, you can plot these points visually for a more interactive experience.