Sketch the curve y = (x^2 - 9)(x - 2)

When it comes to curve sketching, there are a number of tests that you can do to find out crucial characteristics about the curve. They are following:

1. Setting x = 0 will give you the points at which the curve hits the y-axis. In our example we have y = (0 - 9)(0 - 2) = 18, hence it hits the y-axis at the point y = 18.

2. Setting y = 0 will give you the points at which the curve hits the x-axis. In our example we have
(x^2 - 9)(x - 2) = 0
(x + 3)(x - 3)(x - 2) = 0
Hence it hits the x-axis at x = -3, 2, 3.

3. Solving the equation dy/dx = 0 will give the x-coordinates of the stationary points of the curve. In our example we have
y = (x2 - 9)(x - 2)
y = x- 9x - 2x2 + 18
dy/dx = 3x2 - 9 - 4x
(Setting dy/dx = 0)
3x2 - 4x - 9 = 0
(This can be solved using the quadratic formula)
x = ( 4 +- (16 - 4*3*(-9))1/2 )/( 2*3 )
x = -1.189 or x = 2.523
Hence there are stationary points here.

4. Finally, it is useful to see how y behaves when x tends to plus infinity and minus infinity. In our example, we can see that as x goes to plus infinity we also have (x- 9) goes to infinity and (x - 2) goes to infinity. As a result, y goes to plus infinity. Also, we can see that as x goes to minus infinity we have (x- 9) goes to plus infinity and (x - 2) goes to minus infinity. As a result, y goes to minus infinity.

From these tests we can gain enough information in order to sketch the curve.

Dan S. A Level Maths tutor, A Level Further Mathematics  tutor

11 months ago

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