find any intercepts of the graph of the given equation. determine whether the graph of the equation possesses symmetry with respect to the x-axis, y-axis, or orgin
|x|+|y|=4
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Your graph is a square with vertices
(4,0), (0,4), (-4,0) and (0,-4)
Those are also your x and y intercepts
The graph has symmetry about the x-axis, and about the y-axis , as well as the origin.
To find the intercepts of the graph of the equation |x| + |y| = 4, we need to determine the points where the graph intersects the x-axis and y-axis.
To find the x-intercepts, we set y = 0 and solve for x. Substituting y = 0 into the equation, we get |x| + |0| = 4, which simplifies to |x| = 4. This equation has two solutions: x = 4 and x = -4. Therefore, the graph intersects the x-axis at the points (4, 0) and (-4, 0).
To find the y-intercepts, we set x = 0 and solve for y. Substituting x = 0 into the equation, we get |0| + |y| = 4, which simplifies to |y| = 4. This equation also has two solutions: y = 4 and y = -4. Therefore, the graph intersects the y-axis at the points (0, 4) and (0, -4).
Now, let's analyze the symmetry of the graph with respect to the x-axis, y-axis, and origin.
Symmetry with respect to the x-axis means that if we reflect the graph across the x-axis, it remains unchanged. To check for symmetry with respect to the x-axis, we need to determine if replacing y by -y in the equation results in an equivalent equation. In this case, |x| + |-y| = 4 simplifies to |x| + |y| = 4, which is the same equation as the original. Therefore, the graph of this equation possesses symmetry with respect to the x-axis.
Symmetry with respect to the y-axis means that if we reflect the graph across the y-axis, it remains unchanged. To check for symmetry with respect to the y-axis, we need to determine if replacing x by -x in the equation results in an equivalent equation. In this case, |-x| + |y| = 4 simplifies to |x| + |y| = 4, which is again the same as the original equation. Therefore, the graph of this equation possesses symmetry with respect to the y-axis.
Symmetry with respect to the origin means that if we reflect the graph across the origin (0,0), it remains unchanged. To check for symmetry with respect to the origin, we need to determine if replacing x by -x and y by -y in the equation results in an equivalent equation. In this case, |-x| + |-y| = 4 simplifies to |x| + |y| = 4, which is identical to the original equation. Therefore, the graph of this equation possesses symmetry with respect to the origin.
In summary, the graph of the equation |x| + |y| = 4 intersects the x-axis at (4,0) and (-4,0), the y-axis at (0,4) and (0,-4), and possesses symmetry with respect to the x-axis, y-axis, and origin.