Problem 52 - 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? $$y=-x^{2}-4 x+4$$

Step-by-Step Solution

Answer

The given quadratic equation has a vertex at (2, -4) and opens downward. Its domain is all real numbers, or (-∞, ∞), and its range is (-∞, -4]. The relation represented by this equation is a function.

Step 1: Determine the vertex

The vertex of the parabola is given by the formula \((-b/2a, f(-b/2a))\), where a and b are coefficients in the quadratic function of the form \(f(x) = ax^2 + bx + c\). Here, a=-1, and b=-4. So, the x-coordinate of the vertex is \(-(-4)/(2*(-1)) = 2\). Substituting \(x = 2\) into the original equation gets the y-coordinate: \(y = -(2)^2 - 4*(2) + 4 = -4\). Thus, the vertex is at (2,-4).

Step 2: Determine the direction the parabola opens

The sign of the coefficient a in the equation of the parabola determines whether the parabola opens upward or downward. Here, since a = -1 and is negative, the parabola opens downward.

Step 3: Determine the relation's domain and range

The domain of a parabola is always all real numbers, so the domain is (-∞, ∞). The range of a parabola that opens downward is all values that are less than or equal to the y-coordinate of the vertex. Since the vertex is at (2, -4), the range of this function is (-∞, -4].

Step 4: Determine if the relation is a function

The vertical line test can check whether a relation is a function. If any vertical line drawn through the graph of the relation crosses the graph at more than one point, the relation is not a function. In the case of a parabola, any vertical line will intersect the graph at most once. Thus the relation is a function.