Question
Give the domain and range of the quadratic function whose graph is described. The vertex is \( (-5,-4) \) and the parabola opens down. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The range of the function is \( (-\infty,-4) \). (Type your answer in interval notation.)
Ask by Hills Hanson. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Domain:** \((-∞, ∞)\)
**Range:** \((-∞, -4]\)
Solution
For the given quadratic function:
- **Domain:** The domain of any quadratic function is all real numbers since a parabola extends infinitely in the horizontal direction.
**Domain:** \((-∞, ∞)\)
- **Range:** Since the parabola opens downward and the vertex is at \((-5, -4)\), the highest point on the graph is \(-4\). Therefore, the range includes all real numbers less than or equal to \(-4\).
**Range:** \((-∞, -4]\)
**Summary:**
- **Domain:** \((-∞, ∞)\)
- **Range:** \((-∞, -4]\)
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Simplify this solution Beyond the Answer
The domain of the quadratic function is all real numbers, expressed as \( (-\infty, \infty) \). This is because a quadratic function can take any real number as input, allowing the graph to span horizontally across the entire x-axis. Since the parabola opens downwards and the vertex is at \( (-5, -4) \), the highest point on the graph is at the vertex, meaning the range of the function includes all values less than or equal to -4. Therefore, the range of the function is written as \( (-\infty, -4] \).
