How do you sketch the graph f(x)=2x^4-26x^2+72?

1 Answer
George C.
Oct 12, 2016

See explanation...

Explanation:

f(x) = 2x^4-26x^2+72

color(white)(f(x)) = 2((x^2)^2-13(x^2)+36)

color(white)(f(x)) = 2(x^2-4)(x^2-9)

color(white)(f(x)) = 2(x-2)(x+2)(x-3)(x+3)

So the graph of this function intercepts the x axis at x=+-2 and x=+-3.

It intercepts the y axis at (0, 72), where it has a local maximum.

Note that f(x) is an even function, symmetric about the y axis.

f'(x) = 8x^3-52x = 2((2x)^2-26)x

So this quartic has local minima at:

x = +-sqrt(26)/2 ~~ +-5.1/2 = +-2.55

We find:

f(+-sqrt(26)/2) = 2(13/2)^2-26(13/2)+72 = 169/2-169+72 = -25/2

So this quartic function is a classic "W" shaped curve, symmetric about the y axis, passing through:

(-3, 0), (-sqrt(26)/2, -25/2), (-2, 0), (0, 72), (2, 0), (sqrt(26)/2, -25/2), (3, 0)

graph{2x^4-26x^2+72 [-10, 10, -20, 80]}

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