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Chapter 3: Problem 45

Give the domain and the range of each quadratic function whose graph is described. The vertex is \((-1,-2)\) and the parabola opens up.

Short Answer

Expert verified
The domain of the given quadratic function is all real numbers, i.e., (-鈭�, 鈭�), and the range is from -2 to infinity, i.e., [-2, 鈭�).

Step by step solution

01

Determine the Domain

The domain of a quadratic function is all real numbers. That is because you can plug in any real number for the x variable in a quadratic function and it will output a y value. Therefore, the domain is expressed as {x 鈭� 鈩潁 or (-鈭�, 鈭�).
02

Determine the Range

The range of a quadratic function depends on whether the graph opens upwards or downwards and the y-coordinate of the vertex of the function. If the graph opens upwards (i.e., the leading coefficient or the coefficient of \(x^2\) is positive), then the range is from the y-coordinate of the vertex to positive infinity (inclusive) because the graph will continue to increase for the rest of its length after reaching this turning point. Here, given the vertex is at (-1,-2) and the parabola opens upwards, the range is {y | y 鈮� -2}, or as an interval (-2, 鈭�).

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