Question

question
determine the domain on which the following function is increasing.
(graph of a parabola opening upwards, with vertex around x = -3, y-intercept at (0, 2) or similar, x-axis from -10 to 10, y-axis from -10 to 10)
answer
(input box with symbols: φ, or, ∪, ∞, <, >, ≤, ≥, ,, (,), (,, (,), all real numbers)

Explanation:

Step1: Identify the vertex's x - coordinate

The graph is a parabola opening upwards (since the coefficient of \( x^{2} \) is positive, as the parabola opens up). The vertex of a parabola \( y = ax^{2}+bx + c \) is at \( x=-\frac{b}{2a} \). From the graph, the vertex (the minimum point) has an x - coordinate of \( x=- 3 \).

Step2: Determine the interval where the function is increasing

For a parabola opening upwards, the function is increasing for all \( x \) - values greater than the x - coordinate of the vertex. So the function is increasing on the interval \( (-3, \infty) \).

Answer:

\( (-3, \infty) \)