Problem 55 - Algebra and Trigonometry

Question

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 \(x=11\)

Step-by-Step Solution

Answer

The equation in standard form of the parabola that has the same shape as \(f(x) = 3x^2\), but with the minimum of 0 at x = 11 is \(f(x) = 3 \cdot (x-11)^2\).

Step 1: Understand the Given Standard Equations of a Parabola

The general equation of a parabola is given by \(f(x) = a \cdot (x-h)^2 + k\), where a, h, and k are constants. The value of a determines the direction and the width of the parabola. If a is greater than zero, the parabola opens upward, and if it is less than zero, the parabola opens downwards. For the functions \(f(x) = 3x^2\) and \(g(x)=-3x^2\), a is either 3 or -3 respectively, meaning they open upward or downward. Here, we need to find a similar parabola that has its minimum at x = 11. Hence, our parabola should open upwards as to have a minimum point. Therefore, a should be positive in our equation.

Step 2: Input the Minimum at x = 11 into the Standard Equation

The minimum point of a parabola equation is where the vertex of the parabola is, given as the point (h, k). It is said that this parabola has a minimum of 0 when x=11. This means that h = 11 and k = 0. Also, since the problem states that the new parabola should have the same shape as \(f(x) = 3x^2\), the value of a should be 3.

Step 3: Formulate the New Equation

Using the determined values of a, h, and k to fill into the standard form of a parabola equation \(f(x) = a \cdot (x-h)^2 + k\), the new equation would be \(f(x) = 3 \cdot (x-11)^2 + 0\), which simplifies to \(f(x) = 3 \cdot (x-11)^2\).