Problem 54 - 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. Maximum \(=-7\) at \(x=5\)

Step-by-Step Solution

Answer

The equation of the parabola in standard form that has the same shape as the graph of \(f(x)=3x^2\) or \(g(x)=-3x^2\), but with a maximum of -7 at x=5 is \(g(x)=-3x^2+30x-82\).

Step 1: Analyzing the given parameters

The problem gives the maximum value of -7 at x=5. This means the vertex of the parabola is (5,-7). Because the parabola has a maximum, it opens downwards, meaning the coefficient a will be negative.

Step 2: Identifying the original function

The original function that matches the direction (opening downwards) is \(g(x)=-3x^2\). This indicates our function will also have a negative coefficient.

Step 3: Applying transformation to the standard parabola

To match the given vertex, we need to shift the graph right by 5 units and down by 7 units. This leads to the new function: \(g(x)=-3(x-5)^2-7\).

Step 4: Simplifying our function

We simplify our expression so that it is in the standard quadratic form. Simplifying gives \(g(x)=-3(x^2-10x+25)-7=-3x^2 + 30x -75 -7=-3x^2+30x-82\).