from the graph of the parabola, determine whether the parabola opens upward or downward, and find the…

Explanation:

Step1: Observe the parabola shape

By looking at the graph, the parabola opens upward as it has a U - shape.

Step2: Identify the vertex

The vertex is the lowest point of the parabola. From the graph, the vertex is at the point $(-1,-8)$.

Step3: Determine the axis of symmetry

The axis of symmetry is a vertical line passing through the vertex. For a parabola with vertex $(h,k)$, the equation of the axis of symmetry is $x = h$. So, the axis of symmetry is $x=-1$.

Step4: Find the range

Since the parabola opens upward and the $y$-coordinate of the vertex is $- 8$, the range is $y\geq - 8$, or in interval notation $[-8,\infty)$.

Step5: Find the maximum/minimum value

The minimum value of the function occurs at the vertex. So the minimum value is $y = - 8$ and there is no maximum value.

Step6: Determine increasing and decreasing intervals

The function is decreasing on the interval $(-\infty,-1)$ and increasing on the interval $(-1,\infty)$.

Answer:

The parabola opens upward. Vertex: $(-1,-8)$ Axis of symmetry: $x = - 1$ Range: $[-8,\infty)$ Minimum value: $y=-8$, no maximum value Decreasing interval: $(-\infty,-1)$ Increasing interval: $(-1,\infty)$