To find the absolute maximum and minimum values of the quadratic function shown in the graph, let's analyze the shape and position of the parabola.

Analysis of the Parabola

1. Orientation: The parabola opens downward, indicating that it has an absolute maximum but no absolute minimum, as it extends indefinitely downwards.
2. Vertex: The vertex is at the highest point of the parabola, which is the absolute maximum. Observing the graph, the vertex appears to be at the point $(2, 4)$.

Absolute Maximum and Minimum

• Absolute Maximum: The highest point is the vertex at $y = 4$.
• Absolute Minimum: Since the parabola opens downward and extends infinitely in the negative direction, there is no absolute minimum.

Conclusion

• Absolute Minimum: DNE (Does Not Exist)
• Absolute Maximum: $4$

Would you like more details on why the minimum does not exist or have any questions about this solution?

Related Questions

1. How can we determine if a parabola has an absolute maximum or minimum just by looking at its equation?
2. What does the orientation of the parabola (upward or downward) tell us about the maximum and minimum values?
3. How can we use the vertex form of a quadratic function to identify the maximum or minimum?
4. How do we find the vertex of a parabola given in standard form?
5. How do absolute maximum and minimum values relate to local maxima and minima?

Tip

For any quadratic function $ax^2 + bx + c$, the vertex (where the maximum or minimum occurs) is given by $x = -\frac{b}{2a}$.