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

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2 Answers

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.

 

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=x1/3. 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: = (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 = y1/3. 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 xy3. 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 -xy3, 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.

Comments

Ricardo, you may want to revise your solution. Particularly toward the end where you say that the function is not symmetric about the origin. In fact, this function is symmetric about the origin. You are correct that after replacing x and y with -x and -y respectively, that it results in -x = -y3. But this is equivalent to the original (multiply both sides by -1), and this is where I think you have made the mistake.