Problem 51 - College Algebra

Question

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}+4 x-3$$

Step-by-Step Solution

Answer

The domain of the relation is \(\{-\infty, \infty\}\) and the range is \(-\infty, 1]\). Yes, the relation is a function because it passes the vertical line test.

Step 1: Convert to vertex form

Rewrite the equation in vertex form by completing the square. The given equation \(y = -x^{2} + 4x - 3\), should be transformed using the following steps: \n\n1. Group the x-terms together \(y = -(x^2 - 4x) - 3\)\n\n2. To complete the square inside the brackets, take half the coefficient of x, square it and add it inside the brackets, and subtract it outside, adjusted for the coefficient inside the brackets \(y = -(x^2 - 4x + 4) - 3 + 4\)\n\n3. Now our equation in vertex form is \(y = -(x-2)^2 + 1\)

Step 2: Determine the vertex and direction of the parabola

In the equation \(y = -(x-2)^2 + 1\), the vertex is \((2,1)\), and since the coefficient of \((x-2)^2\) is negative, it means that the parabola opens downwards.

Step 3: Determine the domain and range of the relation

The domain of a quadratic function is all the set of real numbers. So the domain is \(\{-\infty, \infty\}\). The range of the function is determined by the vertex and the direction in which the parabola opens. Since the parabola opens downwards, the maximum value will be at the vertex, meaning that the range is \(-\infty,1]\).

Step 4: Establish if the relation is a function

A relation is a function if every x value has exactly one corresponding y value. This is also known as the vertical line test. For a parabola, any vertical line drawn through it will intersect the graph only once, so this relation is a function.