Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ |x|+|y|=4 $$

Short answer

The graph of \(|x| + |y| = 4\) is a square centered at the origin, with vertex points \((4,0), (-4,0), (0,4), (0,-4)\).

Step-by-step solution

Identify the symmetry

The equation \(|x| + |y| = 4\) is symmetric with respect to both the x-axis, y-axis, and the origin. This symmetry arises because the absolute values of \(x\) and \(y\) are unaffected by changes in the sign of \(x\) or \(y\). This means the graph will have a uniform shape in the first, second, third, and fourth quadrants.

Determine the intercepts

To find the x-intercept, set \(y = 0\) and solve:\[|x| + |y| = |x| + |0| = 4\]\(|x| = 4\), hence \(x = 4\) or \(x = -4\). Thus, the x-intercepts are \((4, 0)\) and \((-4, 0)\).To find the y-intercept, set \(x = 0\) and solve:\[|x| + |y| = |0| + |y| = 4\] \(|y| = 4\), hence \(y = 4\) or \(y = -4\). Thus, the y-intercepts are \((0, 4)\) and \((0, -4)\).

Plot key points

Using the intercepts found, plot the points \((4, 0)\), \((-4, 0)\), \((0, 4)\), and \((0, -4)\) on the coordinate plane. These points are crucial for shaping the graph in each quadrant.

Sketch the complete graph

Connecting the points \((4, 0)\), \((0, 4)\), \((-4, 0)\), and \((0, -4)\) will form a square (or diamond) centered at the origin. Be sure that the edges of the square are linear and connect through these intercepts. The symmetry of the equation ensures the shape remains uniform across all four quadrants.

Key concepts

Symmetry in Graphs

Symmetry in graphs refers to a situation where a particular shape or pattern repeats itself across an axis, or around a central point. In the case of the equation \(|x| + |y| = 4\), it exhibits symmetry with respect to the x-axis, y-axis, and the origin. This is due to the properties of absolute value: whether \(x\) or \(y\) is positive or negative, the absolute value remains the same.
This type of symmetry implies that changes in signs of \(x\) or \(y\) do not affect the overall graph, creating a uniform shape.
When graphing such an equation, expect the diagram to look the same in all four quadrants of the coordinate plane, often resulting in familiar shapes – such as circles, squares, or diamonds – depending on the equation's complexity.
Understanding this concept can greatly reduce the complexity of plotting as you only need to understand one section, and then mirror it appropriately.

X-intercepts and Y-intercepts

Finding the x-intercepts and y-intercepts of a graph is a foundational tool in graphing techniques. These intercepts are points where the graph crosses the x-axis and y-axis, respectively.
To identify the x-intercepts in the equation \(|x| + |y| = 4\), set \(y = 0\). This simplifies the equation to \(|x| = 4\), yielding intercepts at \((4, 0)\) and \((-4, 0)\).
For the y-intercepts, set \(x = 0\). The equation becomes \(|y| = 4\), resulting in points \((0, 4)\) and \((0, -4)\).
These intercepts not only provide specific points to plot but help define the boundaries and shape of the graph. In symmetrical graphs like this, they are instrumental in ensuring the symmetry is correctly represented.

Graphing Techniques

Graphing techniques are strategies used to simplify the plotting process and make it more precise. When plotting \(|x| + |y| = 4\), it's essential to start by marking the intercepts on the coordinate plane.
Once the intercepts \((4, 0)\), \((-4, 0)\), \((0, 4)\), and \((0, -4)\) are identified, the next step is connecting these points in a logical order. For absolute value graphs, expect straight lines, forming a geometric shape like a diamond.
Make sure your lines are precise and span through the origin, reflecting the symmetry we've identified.
This straightforward technique creates a complete and accurate graph quite efficiently, useful for equations involving absolute values or other symmetrical properties.

Coordinate Plane

The coordinate plane is a two-dimensional space where graphs are plotted. It consists of two perpendicular lines: the horizontal x-axis and the vertical y-axis. Both axes intersect at the origin (0,0).
This plane allows for the representation of equations like \(|x| + |y| = 4\), where each variable corresponds to an axis. The quadrants, labeled I to IV starting from the top right and moving counterclockwise, help in understanding how the graph reflects its symmetry.
Plotting on a coordinate plane requires understanding its scale and axis divisions, ensuring intercepts and other critical points are accurately represented.
This foundational system is crucial in mathematics for representing vast amounts of data and understanding the relationships between variables.