QuestionDetermine symmetry of the following equations: 17. $y=x^{2}+6$ , 19. $x=y^{2}+6$ , 18. $y=x^{2}-2$ , 21. $y^{2}=x^{2}+6$ , 20. $x=y^{2}-2$ , 23. $y=2x+3$ , 22. $y^{2}=x^{2}-2$ , 25. $x^{2}-y^{3}=2$ , 24. $y=2x+5$ , 27. $x^{2}+y^{2}=100$ , 26. $x^{3}-y^{2}=5$ , 28. $x^{2}+y^{2}=49$ .

Studdy Solution

STEP 1

Assumptions1. The equations provided are in the form of $y=f(x)$ or $x=f(y)$ or $f(x,y)=0$ . . We are to determine the symmetry of the graph of each equation with respect to the $y$ -axis, the $x$ -axis, the origin, more than one of these, or none of these.

STEP 2

To check for symmetry about the $y$ -axis, we replace $x$ with $-x$ in the equation and see if we get the same equation back.

STEP 3

To check for symmetry about the $x$ -axis, we replace $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 4

To check for symmetry about the origin, we replace $x$ with $-x$ and $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 5

Let's start with the first equation, $y=x^{2}+$ . Replace $x$ with $-x$ .
$y=(-x)^{2}+$

STEP 6

implify the equation.
$y=x^{2}+6$

STEP 7

Since we got the same equation back, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis.

STEP 8

Now, let's check for symmetry about the $x$ -axis. Replace $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=x^{2}+6$

STEP 9

This is not the same as the original equation, so the graph of the equation $y=x^{2}+6$ is not symmetric with respect to the $x$ -axis.

STEP 10

Finally, let's check for symmetry about the origin. Replace $x$ with $-x$ and $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=(-x)^{2}+6$

STEP 11

This is not the same as the original equation, so the graph of the equation $y=x^{}+6$ is not symmetric with respect to the origin.

STEP 12

Therefore, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis only.

STEP 13

Repeat steps5 to12 for each of the remaining equations to determine their symmetry.

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QuestionDetermine symmetry of the following equations: 17. $y=x^{2}+6$ , 19. $x=y^{2}+6$ , 18. $y=x^{2}-2$ , 21. $y^{2}=x^{2}+6$ , 20. $x=y^{2}-2$ , 23. $y=2x+3$ , 22. $y^{2}=x^{2}-2$ , 25. $x^{2}-y^{3}=2$ , 24. $y=2x+5$ , 27. $x^{2}+y^{2}=100$ , 26. $x^{3}-y^{2}=5$ , 28. $x^{2}+y^{2}=49$ .

Studdy Solution

STEP 1

Assumptions1. The equations provided are in the form of $y=f(x)$ or $x=f(y)$ or $f(x,y)=0$ . . We are to determine the symmetry of the graph of each equation with respect to the $y$ -axis, the $x$ -axis, the origin, more than one of these, or none of these.

STEP 2

To check for symmetry about the $y$ -axis, we replace $x$ with $-x$ in the equation and see if we get the same equation back.

STEP 3

To check for symmetry about the $x$ -axis, we replace $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 4

To check for symmetry about the origin, we replace $x$ with $-x$ and $y$ with $-y$ in the equation and see if we get the same equation back.

STEP 5

Let's start with the first equation, $y=x^{2}+$ . Replace $x$ with $-x$ .
$y=(-x)^{2}+$

STEP 6

implify the equation.
$y=x^{2}+6$

STEP 7

Since we got the same equation back, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis.

STEP 8

Now, let's check for symmetry about the $x$ -axis. Replace $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=x^{2}+6$

STEP 9

This is not the same as the original equation, so the graph of the equation $y=x^{2}+6$ is not symmetric with respect to the $x$ -axis.

STEP 10

Finally, let's check for symmetry about the origin. Replace $x$ with $-x$ and $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=(-x)^{2}+6$

STEP 11

This is not the same as the original equation, so the graph of the equation $y=x^{}+6$ is not symmetric with respect to the origin.

STEP 12

Therefore, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis only.

STEP 13

Repeat steps5 to12 for each of the remaining equations to determine their symmetry.

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Studdy Solution

STEP 1

Assumptions1. The equations provided are in the form of y=f(x)y=f(x)y=f(x) or x=f(y)x=f(y)x=f(y) or f(x,y)=0f(x,y)=0f(x,y)=0. . We are to determine the symmetry of the graph of each equation with respect to the yyy-axis, the xxx-axis, the origin, more than one of these, or none of these.

STEP 2

To check for symmetry about the yyy-axis, we replace xxx with −x-x−x in the equation and see if we get the same equation back.

STEP 3

To check for symmetry about the xxx-axis, we replace yyy with −y-y−y in the equation and see if we get the same equation back.

STEP 4

To check for symmetry about the origin, we replace xxx with −x-x−x and yyy with −y-y−y in the equation and see if we get the same equation back.

STEP 5

Let's start with the first equation, y=x2+y=x^{2}+y=x2+. Replace xxx with −x-x−x.y=(−x)2+y=(-x)^{2}+y=(−x)2+

STEP 6

implify the equation.y=x2+6y=x^{2}+6y=x2+6

STEP 7

Since we got the same equation back, the graph of the equation y=x2+6y=x^{2}+6y=x2+6 is symmetric with respect to the yyy-axis.

STEP 8

Now, let's check for symmetry about the xxx-axis. Replace yyy with −y-y−y in the equation y=x2+6y=x^{2}+6y=x2+6.−y=x2+6-y=x^{2}+6−y=x2+6

STEP 9

This is not the same as the original equation, so the graph of the equation y=x2+6y=x^{2}+6y=x2+6 is not symmetric with respect to the xxx-axis.

STEP 10

Finally, let's check for symmetry about the origin. Replace xxx with −x-x−x and yyy with −y-y−y in the equation y=x2+6y=x^{2}+6y=x2+6.−y=(−x)2+6-y=(-x)^{2}+6−y=(−x)2+6

STEP 11

This is not the same as the original equation, so the graph of the equation y=x+6y=x^{}+6y=x+6 is not symmetric with respect to the origin.

STEP 12

Therefore, the graph of the equation y=x2+6y=x^{2}+6y=x2+6 is symmetric with respect to the yyy-axis only.

STEP 13

Repeat steps5 to12 for each of the remaining equations to determine their symmetry.

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Assumptions1. The equations provided are in the form of $y=f(x)$ or $x=f(y)$ or $f(x,y)=0$ . . We are to determine the symmetry of the graph of each equation with respect to the $y$ -axis, the $x$ -axis, the origin, more than one of these, or none of these.

To check for symmetry about the $y$ -axis, we replace $x$ with $-x$ in the equation and see if we get the same equation back.

To check for symmetry about the $x$ -axis, we replace $y$ with $-y$ in the equation and see if we get the same equation back.

To check for symmetry about the origin, we replace $x$ with $-x$ and $y$ with $-y$ in the equation and see if we get the same equation back.

Let's start with the first equation, $y=x^{2}+$ . Replace $x$ with $-x$ .
$y=(-x)^{2}+$

implify the equation.
$y=x^{2}+6$

Since we got the same equation back, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis.

Now, let's check for symmetry about the $x$ -axis. Replace $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=x^{2}+6$

This is not the same as the original equation, so the graph of the equation $y=x^{2}+6$ is not symmetric with respect to the $x$ -axis.

Finally, let's check for symmetry about the origin. Replace $x$ with $-x$ and $y$ with $-y$ in the equation $y=x^{2}+6$ .
$-y=(-x)^{2}+6$

This is not the same as the original equation, so the graph of the equation $y=x^{}+6$ is not symmetric with respect to the origin.

Therefore, the graph of the equation $y=x^{2}+6$ is symmetric with respect to the $y$ -axis only.