Problem 38 Graph the equations by plotting ... [FREE SOLUTION] (2024)

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Plotting points is a fundamental skill in graphing equations. It involves placing specific coordinates on a graph based on given equations. Let's break this down. To start, you'll need:

• The equation to graph
• A list of points derived from substituting values into the equation
• A graph paper or a graphing tool

For example, in the equation \(|x| + y = 3\), you substitute positive and negative \(\text{values of } x\text { to } find y:\). This gives you points you can plot: \((0, 3), (1, 2), (2, 1), (3, 0), (-1, 2), (-2, 1), (-3, 0)\). After plotting these points, connect them to form the shape of the graph. Make sure each point is accurately placed using the coordinates, and draw straight lines to connect them. Plotting points is essential for understanding how the equation translates visually.

The Cartesian coordinate system is a two-dimensional plane divided by a horizontal line \(x-axis\) and a vertical line \(y-axis\). This system helps us to visually represent mathematical equations. Every point on the plane is defined by an ordered pair \((x, y)\). Here’s how it works:

• Each point is determined by how far along the x-axis and y-axis it is.
• The origin \( (0, 0)\) is the point where the x-axis and y-axis intersect.
• Positive values of x are to the right of the origin, and negative values are to the left.
• Positive values of y are above the origin, and negative values are below.

To plot the equation \( |x| + y = 3\), you’ll use the Cartesian plane. Place each calculated point \((0, 3), (1, 2), (2, 1), (3, 0), (-1, 2), (-2, 1), (-3, 0)\)while keeping track of their exact locations in relation to the origin. Understanding this system is crucial for accurate graphing.

An absolute value graph represents equations that include absolute value expressions, like \(|x|\text{ in } |x| + y = 3\). The absolute value \(|x|\) makes sure that only non-negative values of x are represented. Here’s the step-by-step:

• When x is positive or zero, \(|x| = x\).
• When x is negative, \(|x| = -x\).
• In essence, \(|x|\) reflects all negative x values as positive, creating symmetry.

For this equation \(|x| + y = 3\), this symmetry manifests in the plotted points: \((1, 2)\) mirrors \((-1, 2)\), \((2, 1)\) mirrors \((-2, 1)\). Plotting these on the coordinate system forms a 'V' shape. Absolute value graphs are unique due to this V-shaped symmetry and are extremely helpful in visualizing how equations behave differently with positive and negative inputs.