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Chapter 13: Problem 78

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

Short Answer

Expert verified
The vertex of the parabola is (2, -4). Because the parabola opens downwards, its domain is all real numbers and its range is \(y \leq -4\). The relation is a function, as every \(x\) value maps to exactly one \(y\) value.

Step by step solution

01

Determine the vertex

The vertex from an equation in the form of \(y = ax^2 + bx + c \) is given by \((-\frac{b}{2a}, f(-\frac{b}{2a}))\), where \(f(x)\) is the equation of the parabola. Here, \(a = -1, b = -4\), so the vertex is \(\left(\frac{-(-4)}{2*-1}, -\left(\frac{-(-4)}{2*-1}\right)^2 - 4*\left(\frac{-(-4)}{2*-1}\right) + 4\right)\) = (2, -4).
02

Determine the direction of the opening

Since the coefficient \(a\) of \(x^2\) in the given equation is negative, the parabola opens downward. This is key in finding the range of the function.
03

Determine the domain and range

For the domain of a parabola, all real values of x are possible, so the domain is all real numbers. For the range, because the parabola opens downwards and the highest point is the vertex, any value less than or equal to the y-coordinate of the vertex is in the range. Here, the range is \(y \leq -4\).
04

Determine if the relation is a function

The given relation is indeed a function because it is a parabola. Any value of \(x\) will return a single \(y\) value, hence, the relation is also a function.

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Most popular questions from this chapter

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=y^{2}+6 y+5$$

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