What is the vertex form of #y= x^2-x-20 #?

Answer 1

#(1/2,-81/4)#

The vertex, also known as the turning point, is the function's relative extreme point, and it happens at the point where the function's derivative is zero.

That is, when #dy/dx=0#

ie when #2x-1=0# which implies #x=1/2#.

The corresponding y values is then #y(1/2)=(1/2)^2-1/2-20=-81/4#.

Since the coefficient of #x^2# is #1>0#, it implies the arms of the corresponding parabola graph of this quadratic function go up and hence the relative extremum is a relative (and in fact an absolute) minimum. One could also check this by showing that the second derivative #(d^2y)/(dx^2)|_(x=1/2)=2>0#.

For completeness, the corresponding graph is provided.

plot{x^2-x-20 [-11.95, 39.39, -22.35, 3.28]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.