Problem 77 - College Algebra

Question

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Step-by-Step Solution

Answer

The process to find the vertex of a parabola with equation in standard form involves identifying the coefficients of the equation, then using these in the formulas for the vertex's coordinates. For the example \(y = 2x^2 + 6x + 3\), the vertex is at the point \(-1.5, -1.5\).

Step 1: Identify the coefficients

Given a standard form equation \(y = ax^2 + bx + c\), identify the coefficients a, b, and c. For example, in the equation \(y = 2x^2 + 6x + 3\), a=2, b=6, and c=3.

Step 2: Find the x-coordinate of the vertex

The x-coordinate of the vertex (h) is given by the formula \(h = -b/(2a)\). Plug the identified coefficients into this formula to find the x-coordinate of the vertex. For our example, \(h = -b/(2a) = -6/(2*2) = -6/4 = -1.5\).

Step 3: Find the y-coordinate of the vertex

The y-coordinate of the vertex (k) is given by the formula \(k = c - b^2/(4a)\). Plug the identified coefficients into this formula to find the y-coordinate of the vertex. For our example, \(k = c - b^2/(4a) = 3 - (6)^2/(4*2) = 3 - 36/8 = 3 - 4.5 = -1.5\).

Step 4: Write down the vertex

Having found both h and k, the vertex of the parabola is therefore at the point \(h,k\). For our example, the vertex is at the point \(-1.5, -1.5\).