Question

Determine algebraically whether the graphs of the equations are symmetric to x-axis, y-axis, origin, or none of these.
A. \[y = - |x| + 5\]
B. \[xy = 4\]
C. \[x + {y^2} = 8\]

Answer 1

Hint: Here we will apply the tests for symmetricity of a curve about x axis, y axis and origin and then state the given curves as symmetric about x axis, y axis or origin.
The test for x axis:-
Replace y with (-y) then simplify the equation. If the resulting equation is equal to the original equation then the graph is symmetrical about the x-axis.
The test for y axis: -
Replace x with (-x) then simplify the equation. If the resulting equation is equal to the original equation then the graph is symmetrical about the y-axis.
The test for origin:-
Replace y with (-y) and x with (-x) then simplify the equation. If the resulting equation is equal to the original equation then the graph is symmetrical about the origin.

Complete step-by-step answer:
Let us first consider part A.
A. \[y = - |x| + 5\]
The test for x axis:-
Replace y with (-y) then simplify the equation. If the resulting equation is equal to the original equation
then the graph is symmetrical about the x-axis.
Hence, replacing y with (-y) we get:-
\[ - y = - |x| + 5\]
Now since it is not equal to the original equation, therefore it is not symmetric about the x axis.
Test for y axis: -
Replace x with (-x) then simplify the equation. If the resulting equation is equal to the original equation
then the graph is symmetrical about the y-axis.
Hence, replacing x with (-x) we get:-
\[y = - | - x| + 5\]
Simplifying it we get:-
\[y = - |x| + 5\]
Here since we get the same equation as the original equation
Hence, it is symmetric about y axis.
Test for origin:-
Replace y with (-y) and x with (-x) then simplify the equation. If the resulting equation is equal to the original equation then the graph is symmetrical about the origin.
Hence, replacing y with (-y) and x with (-x) we get:-
\[ - y = - | - x| + 5\]
Simplifying it we get:-
\[ - y = - |x| + 5\]
Now since it is not equal to the original equation, therefore it is not symmetric about origin.
Now let us consider part B
B. \[xy = 4\]
The test for x axis:-
Replace y with (-y) then simplify the equation. If the resulting equation is equal to the original equation
then the graph is symmetrical about the x-axis.
Hence, replacing y with (-y) we get:-
\[x\left( { - y} \right) = 4\]
Simplifying it we get:-
\[ - xy = 4\]
Now since it is not equal to the original equation, therefore it is not symmetric about the x axis.
Test for y axis: -
Replace x with (-x) then simplify the equation. If the resulting equation is equal to the original equation
then the graph is symmetrical about the y-axis.
Hence, replacing x with (-x) we get:-
\[\left( { - x} \right)y = 4\]
Simplifying it we get:-
\[ - xy = 4\]
Now since it is not equal to the original equation, therefore it is not symmetric about y axis.
Test for origin:-
Replace y with (-y) and x with (-x) then simplify the equation. If the resulting equation is equal to the
original equation then the graph is symmetrical about the origin.
Hence, replacing y with (-y) and x with (-x) we get:-
\[\left( { - x} \right)\left( { - y} \right) = 4\]
Simplifying it we get:-
\[xy = 4\]
Here since we get the same equation as the original equation
Hence, it is symmetric about origin.
Now let us consider part C
C. \[x + {y^2} = 8\]
The test for x axis:-
Replace y with (-y) then simplify the equation. If the resulting equation is equal to the original equation
then the graph is symmetrical about the x-axis.
Hence, replacing y with (-y) we get:-
\[x + {\left( { - y} \right)^2} = 8\]
Simplifying it further we get:-
\[x + {y^2} = 8\]
Here since we get the same equation as the original equation
Hence, it is symmetric about the x axis.
Test for y axis: -
Replace x with (-x) then simplify the equation. If the resulting equation is equal to the original equation then the graph is symmetrical about the y-axis.
Hence, replacing x with (-x) we get:-
\[ - x + {y^2} = 8\]
Now since it is not equal to the original equation, therefore it is not symmetric about y axis.
Test for origin:-
Replace y with (-y) and x with (-x) then simplify the equation. If the resulting equation is equal to the
original equation then the graph is symmetrical about the origin.
Hence, replacing y with (-y) and x with (-x) we get:-
\[ - x + {\left( { - y} \right)^2} = 8\]
Simplifying it we get:-
\[ - x + {y^2} = 8\]
Now since it is not equal to the original equation, therefore it is not symmetric about origin.

Note: Students should take a note that the modulus of any quantity is always positive and also that if a curve is symmetric about both x axis and y axis or none of them then it is symmetric about the origin.