I need help solving the two problems below. Thanks
For each equation, determine whether its graph is symmetric with respect to the -axis, the -axis, and the origin.
Check all symmetries that apply.
1. y=-�ã(4-x^(2))
2. 34x^(2)+12y^(2)=18
thats supposed to be a neg in front of the 4 in problem 1.
To determine the symmetry of the graphs for the given equations, we need to check if they are symmetric with respect to the x-axis, y-axis, and the origin. Let's go through each equation one by one:
1. Equation: y = -sqrt(4 - x^2)
For the first equation, which is a function involving a square root, we can start by analyzing the power of x in the equation. Here, the power of x is 2, which means the graph will be a parabola.
Now, since the coefficient of x^2 is negative (-1), we know that the parabola will open downward. Thus, the graph is not symmetric with respect to the x-axis.
Next, let's check the symmetry with respect to the y-axis. For this, we substitute x with -x and see if the equation stays the same. Replacing x with -x in the given equation, we have y = -sqrt(4 - (-x)^2), which simplifies to y = -sqrt(4 - x^2). Since the equation remains the same, the graph is symmetric with respect to the y-axis.
Lastly, to check for symmetry with respect to the origin, we substitute both x and y with their negative counterparts. Replacing x with -x and y with -y in the given equation, we get -y = -sqrt(4 - (-x)^2), which simplifies to y = sqrt(4 - x^2). Since the equation is not the same as the original, the graph is not symmetric with respect to the origin.
Therefore, for the first equation, it is symmetric with respect to the y-axis but not symmetric with respect to the x-axis or the origin.
2. Equation: 34x^2 + 12y^2 = 18
For the second equation, we can determine the symmetry by analyzing the power of both x and y. Here, both x and y have a power of 2, indicating that the graph will be an ellipse.
Let's start with the x-axis symmetry test. Replacing y with -y in the given equation, we have 34x^2 + 12(-y)^2 = 18, which simplifies to 34x^2 + 12y^2 = 18. Since the equation is the same as the original, the graph is symmetric with respect to the x-axis.
Moving on to the y-axis symmetry test, replacing x with -x in the given equation, we have 34(-x)^2 + 12y^2 = 18, which simplifies to 34x^2 + 12y^2 = 18. Again, the equation remains unchanged, confirming that the graph is symmetric with respect to the y-axis.
Lastly, for the origin symmetry test, we substitute both x and y with their negative counterparts. Replacing x with -x and y with -y in the given equation, we get 34(-x)^2 + 12(-y)^2 = 18, which simplifies to 34x^2 + 12y^2 = 18. As the equation remains the same, the graph is symmetric with respect to the origin.
Therefore, for the second equation, it is symmetric with respect to the x-axis, y-axis, and the origin.