How to Graph y = 9/x² Without a Calculator?

[Integral0]

(1) How do you graph the equation y = 9 divided by x^2 w/o using a calculator and besides plotting points?

(2) If you replace x with y and y with x, can you reflect the equation 9/x^2 ?

thanks

 

[HallsofIvy]

You can't graph an equation (other than a linear equation) exactly without plotting individual points.

You can get a pretty good approximation by noting that, since x² is always positive, y= 9/x² is above the y-axis and symmetric about the x axis. Of course, for x close to 0, y will be very large: the y-axis is a vertical asymptote. For x very large, y is close to 0: the x-axis is a horizontal asymptote.

(2) If you replace x with y and y with x, can you reflect the equation 9/x^2 ?

I presume you mean changing y= 9/x² into x= 9/y². Yes, that will "reflect" the graph through the line y= x.

 

[M98Ranger]

(1) To graph the equation y = 9 divided by x^2 without using a calculator and besides plotting points, we can use the rules of transformations. The equation y = 9 divided by x^2 can be rewritten as y = 9 * (1/x^2). This allows us to see that the graph will be a reflection of the graph of y = 9 * x^2 about the y-axis.

To graph y = 9 * x^2, we can start with a basic parabola with its vertex at (0,0) and points on either side such as (-1,9) and (1,9). Then, we can reflect these points about the y-axis to get the corresponding points for y = 9 divided by x^2. This will give us a graph that looks like a sideways "U" with its vertex at (0,0) and points on either side such as (-1,1/9) and (1,1/9).

(2) Yes, if we replace x with y and y with x, we can reflect the equation 9/x^2. The resulting equation would be x = 9/y^2. This is because when we reflect a graph about the line y = x, the x and y coordinates switch places. So, the graph of y = 9/x^2 would be the same as the graph of x = 9/y^2, just rotated 90 degrees counterclockwise.