For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin.
4y^4+24x^2=28 and y=1/x^2+5
Check all symmetries that apply
x-axis,y-axis,origin or non of the above.
You need parentheses for the 2nd equation.
Is this,
y = 1/(x^2 + 5)
Or,
y = 1/(x^2) + 5
Sorry y=1/(x^2+5)
4y^4 + 24x^2 = 28
Symmetric to x-axis if an equivalent equation is obtained when y is replaced by -y.
4y^4 + 24x^2 = 28
4(-y)^4 + 24x^2 = 28
4y^4 + 24x^2 = 28
Since, the two equations are equivalent, the graph is symmetric to the x-axis.
Symmetric to y-axis if an equivalent equation is obtained when x is replaced by -x.
4y^4 + 24x^2 = 28
4y^4 + 24(-x)^2 = 28
4y^4 + 24x^2 = 28
Since, the two equations are equivalent, the graph is symmetric
to the y-axis.
Symmetric to the origin if an equivalent equation is obtained when x is replaced by -x and y is replaced by -y.
4y^4 + 24x^2 = 28
4(-y)^4 + 24(-x)^2 = 28
4y^4 + 24x^2 = 28
Since, the two equations are equivalent, the graph is symmetric
to the origin.
I'll leave the 2nd equation for you to try.
I get that the second equation x-axis. Is that right?
Symmetric to y-axis if an equivalent equation is obtained when x is replaced by -x.
It is symmetric to the y-axis, not x-axis.
Well, let's analyze each equation and see if they have any special symmetries:
Equation 1: 4y^4 + 24x^2 = 28
- Axis of symmetry with respect to the x-axis: Since there is no y term in the equation, we can conclude that the graph is symmetric with respect to the x-axis. So, the answer is yes for symmetry with respect to the x-axis.
- Axis of symmetry with respect to the y-axis: Similarly, since there is no x term in the equation, the graph is symmetric with respect to the y-axis as well. So, the answer is yes for symmetry with respect to the y-axis.
- Axis of symmetry with respect to the origin: In this case, the equation is not symmetric with respect to the origin since both x and y terms are present. So, the answer is no for symmetry with respect to the origin.
Equation 2: y = 1/x^2 + 5
- Axis of symmetry with respect to the x-axis: This equation does not possess any y term, so the graph is not symmetric with respect to the x-axis. Therefore, the answer is no.
- Axis of symmetry with respect to the y-axis: Here, as the equation only has x^2 term, the graph is indeed symmetric with respect to the y-axis. So, the answer is yes for symmetry with respect to the y-axis.
- Axis of symmetry with respect to the origin: Given that both x and y terms are present in this equation, the graph is not symmetric with respect to the origin. Hence, the answer is no.
To summarize:
- Equation 1 has symmetry with respect to the x-axis and the y-axis but not the origin.
- Equation 2 only has symmetry with respect to the y-axis.
Hope this analysis didn't go off the "axis" of your attention span!
To determine whether a graph is symmetric with respect to the x-axis, y-axis, or the origin, we need to analyze the equation's properties.
1. Equation: 4y^4 + 24x^2 = 28
a. X-Axis Symmetry:
To test for x-axis symmetry, we replace y with -y in the equation and simplify:
4(-y)^4 + 24x^2 = 28
This simplifies to:
4y^4 + 24x^2 = 28
Since the resulting equation is the same as the original one, we can conclude that this equation has x-axis symmetry.
b. Y-Axis Symmetry:
To test for y-axis symmetry, we replace x with -x in the equation and simplify:
4y^4 + 24(-x)^2 = 28
This simplifies to:
4y^4 + 24x^2 = 28
Since the resulting equation is the same as the original one, we can conclude that this equation has y-axis symmetry.
c. Origin Symmetry:
To test for origin symmetry, we replace x with -x and y with -y in the equation and simplify:
4(-y)^4 + 24(-x)^2 = 28
This simplifies to:
4y^4 + 24x^2 = 28
Again, the resulting equation is the same as the original, so we can conclude that this equation has origin symmetry.
Therefore, the graph of the equation 4y^4 + 24x^2 = 28 is symmetric with respect to the x-axis, y-axis, and the origin.
2. Equation: y = 1/x^2 + 5
a. X-Axis Symmetry:
To test for x-axis symmetry, we replace y with -y in the equation and simplify:
-y = 1/x^2 + 5
However, after simplifying, we see that the resulting equation is not the same as the original one. Therefore, this equation does not have x-axis symmetry.
b. Y-Axis Symmetry:
To test for y-axis symmetry, we replace x with -x in the equation and simplify:
y = 1/(-x)^2 + 5
This simplifies to:
y = 1/x^2 + 5
Since the resulting equation is the same as the original one, we can conclude that this equation has y-axis symmetry.
c. Origin Symmetry:
To test for origin symmetry, we replace x with -x and y with -y in the equation and simplify:
-y = 1/(-x)^2 + 5
After simplifying, we see that the resulting equation is not the same as the original one. Therefore, this equation does not have origin symmetry.
Therefore, the graph of the equation y = 1/x^2 + 5 is symmetric with respect to the y-axis, but it does not have x-axis or origin symmetry.