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Chapter 3: 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 in standard form is \(y = 3x^2 - 66x + 363\).

Step by step solution

01

Identify the values for the vertex form

Given that the minimum value of the parabola is 0 at \(x = 11\), we understand that \((h, k)\) for this parabola is \((11, 0)\).
02

Determine the a value

Given that the parabola has the same shape as \(f(x) = 3x^2\), we understand that the value of \(a\) is 3 as the shape of a parabola is determined by \(\(a\)\) in the equation \(y = a(x-h)^2 + k\).
03

Write the vertex form

Substitute \(a\), \(h\), and \(k\) into the vertex form equation. Hence, the equation becomes \(y = 3(x - 11)^2 + 0\).
04

Convert to the standard form

Expand the equation to receive the standard form. The standard form is then computed as follows: \(y = 3 (x^2 - 22x + 121)\). This expands to \(y = 3x^2 - 66x + 363\).

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