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Chapter 2: Problem 55

Write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)=3 x^{2}\) or \(g(x)=-3 x^{2},\) but with the given maximum or minimum. Minimum \(=0\) at \(x=11\)

Short Answer

Expert verified
The equation for the parabola that meets the given conditions is \(f(x) = 3(x-11)^2\).

Step by step solution

01

Analyze the given functions and the transformations

Given are two functions \(f(x) = 3x^2\) which opens upwards and \(g(x) = -3x^2\) which opens downwards. We have a minimum constraint at \(x=11\), which means our parabola should open upwards like \(f(x)=3x^2\).
02

Formulate the necessary transformation

To achieve a horizontal shift to bring the minimum at \(x=11\), the form of the parabola will be \(f(x) = a(x-h)^2 + k\), where \(h = 11\) and \(k = 0\), and \(a\) defines the opening direction and the width of the parabola.
03

Construct the final equation

Insert the known values to get the final function. The function maintains the same shape as \(f(x) = 3x^2\) (opens upwards, has the same width), it is shifted \(11\) units to the right along \(x\)-axis, and not shifted vertically since \(k = 0\). Hence the final equation is \(f(x) = 3(x - 11)^2 + 0 = 3(x-11)^2\).

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