How do I plot y=x^2-1?

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To plot any graph, you need to find out where the curve cuts the x and y-axis, and where it's stationary points are, aswell as what type of stationary point you have.

Here are the steps:

1) Let x=0; find where the curve passes the y axis.

   Here, y =-1. This means the curve passes the y axis at (0,-1)

2) Let y=0; find where the curve passes the x axis.

   Here we get 0=x2-1 

   So  x2=1 

   The square roots of 1 are: 1 and -1. (1x1=1, -1x-1=1) 

  This means the curve crosses the x axis as (-1, 0) and (1,0).

Remember the order of this equation (the highest power to which a x is raised) gives the number of times the curve crosses the x axis, here it is two. (This is the same for the y axis. Here, there order of y is 1, so the curve passes through the y axis 1 time.


3) Differentiate the equation with respect to x and make this equal to 0; find the stationary points.

   dy/dx =2x=0

   This tells us there is a stationary point at x=0. We know from step 1 that this corresponds to the coordinate (0, -1). 

For the curve to pass through all of these points, it might be clear that the graph is a "sad face", passing through (0, -1), (1, 0) and (-1, 0) with the MINIMUM located at (0,-1)

However, if this is not clear to you, difference the equation again with respect to x. If this gives you a postive answer, you have a minimum, a negative answer means you have a maximum.

4) Differentiate again with respect to x to find the second derivative; does this give you a postive answer?

Here, d2y/dx2= 2. 

Obviously this is a postive number meaning you have a minimum.


Laura P. A Level Maths tutor, GCSE Maths tutor

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