graph, find x and y intercepts and test for symmetry x=y^3

Question

graph, find x and y intercepts and test for symmetry

Answer 1

To Graph:

Create a table of ordered pairs to get a sense of what the graph looks like.

You could begin your table with a few values for x or y.  For instance,

y | -2 -1 0 1 2

x | _ _ _ _ _

Now to complete the table, substitute the values for y into the equation, and solve for x.  For y = -2, substituting -2 for y in the equation gives

x = (-2)^3, and simplifying this gives x = -8.  Now adding this value for x to the table gives

y | -2 -1 0 1 2

x | -8 _ _ _ _

After completing the rest of the table, it should look like this

y | -2 -1 0 1 2

x | -8 -1 0 1 8

and we now can graph the pairs (-8,-2), (-1,-1), (0,0), (1,1), (8,2).  Plot these ordered pairs, and you may get an idea of what the graph looks like.

If more pairs are needed it is simply a matter of repeating the process for other values to get enough pairs to recognize the graph.

x and y-intercepts:

Find the x-intercept by substituting 0 for y in the equation and solving for x.  The resulting ordered pair(s) will have the form (_,0), and they are your x-intercept(s).

Find the y-intercept by substituting 0 for x in the equation and solving for y. The resulting ordered pair(s) will have the form (0,_), and they are your y-intercept(s).

Test for symmetry:

Test for "horizontal" symmetry by replacing 'x' in every instance with '-x'.  If the resulting equation is identical to the original, then the equation has "horizontal" symmetry, also known as "even" symmetry or symmetry over the y-axis.

Test for "vertical" symmetry by replacing 'y' in every instance with '-y'.  If the resulting equation is identical to the original, then the equation has "vertical" symmetry, also known as symmetry over the x-axis.

Test for "odd" symmetry by replacing both 'x' and 'y' with '-x' and '-y', respectively.  If the resulting equation is identical to the original, then the equation has "odd" symmetry, also known as symmetry over the origin.

I hope this helps.  If you need more help, please feel free to ask and I'll be happy to assist you.

 

Answer 2

To solve this problem, it would be preferable to solve this equation for y. Applying the cubic root on both sides, the equation now gives y=x^1/3. Now to find the x- and y- intercepts for this function, two steps must be done. The first step is to substitute 0 in y to find the x-intercept: x = (0)^1/3 = 0. This means that the graph of this function passes through the point (0,0). The next step is to substitute 0 in x to find the y-intercept: 0 = y^1/3. Applying the cubic root on both sides, it is found that y = 0. The function passes through (0,0), so both the x- and y-intercepts are the same. Finally, to use the test of symmetry, three cases have to be considered. First, find out if the function is symmetric about the x-axis. This means that if the variable signs are the same as in the original function after substituting ^-y for y and solving such equation, it is concluded that the function is symmetric about the x-axis. It is found that x = (^-y)³ => x = ^-y³. The solution is not the same as x = y³. So the function is not symmetric about the x-axis. Second, find out if the function is symmetric about the y-axis. This means that if the variable signs are the same as in the original function after substituting ^-x for x and solving such equation, it is concluded that the function is symmetric about the x-axis. It is found that ^-x = y³, which is not the same as the original function. So the function is not symmetric over the x-axis. Finally, find out if the function is symmetric about the origin. For that case, substitute ^-x and ^-y for x and y in the original function. The function has a result of ^-x = ^-y³, which is not the same as the original function, so it is not symmetric about the origin.