What is the vertex form of #y=(x+4)(2x-1)(x-1)#?

Answer 1

Something like:

#f(x) = 2(x+5/6)x^3 - 91/6(x+5/6)+418/27#

The given polynomial is a cubic, not a quadratic. So we cannot reduce it to 'vertex form'.

What is interesting to do is to find a similar concept for cubics.

For quadratics we complete the square, thereby finding the centre of symmetry of the parabola.

For cubics we can make a linear substitution "completing the cube" to find the centre of the cubic curve.

#108 f(x) = 108(x+4)(2x-1)(x-1)#
#color(white)(108f(x)) = 108(2x^3+5x^2-11x+4)#
#color(white)(108f(x)) = 216x^3+540x^2-1188x+432#
#color(white)(108f(x)) = (6x)^3+3(6x)^2(5)+3(6x)(5)^2+(5)^3 -273(6x)-273(5)+1672#
#color(white)(108f(x)) = (6x+5)^3-273(6x+5)+1672#

So:

#f(x) = 1/108 (6x+5)^3 - 91/36(6x+5)+418/27#
#color(white)(f(x)) = 2(x+5/6)^3 - 91/6(x+5/6)+418/27#
From this we can read that the centre of symmetry of the cubic is at #(-5/6, 418/27)# and the multiplier #2# tells us that it is essentially twice as steep as #x^3# (though the linear term subtracts a constant #91/6# from the slope).

graph{(y-(x+4)(2x-1)(x-1))(40(x+5/6)^2+(y-418/27)^2-0.2) = 0 [-6.13, 3.87, -5, 40]}

So in general we can use this method to get a cubic function into the form:

#y = a(x-h)^3+m(x-h)+k#
where #a# is a multiplier indicating the steepness of the cubic compared with #x^3#, #m# is the slope at the centre point and #(h, k)# is the centre point.
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Answer 2

The vertex form of ( y = (x + 4)(2x - 1)(x - 1) ) can be found by expanding and simplifying the expression, and then completing the square. After simplification, the vertex form is:

[ y = 2(x + 1/4)^2 - 9/8 ]

This expression represents the same function in a form that explicitly shows the vertex as (-1/4, -9/8).

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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.

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.

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.

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.

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