Question

1. absolute max: relative max: absolute min: relative min: as ( x \to -infty ), as ( x \to infty ),

Explanation:

Step1: Analyze Absolute Max

The graph is a parabola opening upwards. For an upward - opening parabola, as \(x\) approaches \(\pm\infty\), \(y\) approaches \(\infty\). So, there is no absolute maximum (since the function values keep increasing without bound).

Step2: Analyze Relative Max

A relative maximum is a point where the function changes from increasing to decreasing. For an upward - opening parabola, the function is decreasing before the vertex and increasing after the vertex. There is no point where the function changes from increasing to decreasing, so there is no relative maximum.

Step3: Analyze Absolute Min

The vertex of an upward - opening parabola is the lowest point on the graph. Since the parabola opens upwards, the \(y\) - coordinate of the vertex is the absolute minimum. Let's assume the vertex has a \(y\) - value (we can see from the graph's shape, but since we are analyzing the concept, the absolute minimum is the \(y\) - value of the vertex. However, if we consider the general form, for \(y = ax^{2}+bx + c,a>0\), the absolute minimum is at \(x=-\frac{b}{2a}\) and \(y = f(-\frac{b}{2a})\). But from the graph's behavior, the absolute minimum exists (the vertex) and is the smallest value the function attains.

Step4: Analyze Relative Min

The vertex of the upward - opening parabola is also the relative minimum because around the vertex (in an interval containing the vertex), the vertex is the lowest point. So the relative minimum is the \(y\) - value of the vertex.

Step5: Analyze End - behavior (\(x

ightarrow-\infty\))
For a quadratic function \(y = ax^{2}+bx + c\) with \(a>0\), as \(x
ightarrow-\infty\), \(x^{2}\) becomes very large (positive, since squaring a negative number gives a positive result) and \(ax^{2}\) dominates the other terms. So \(y = ax^{2}+bx + c
ightarrow\infty\) as \(x
ightarrow-\infty\).

Step6: Analyze End - behavior (\(x

ightarrow\infty\))
Similarly, for \(y = ax^{2}+bx + c\) with \(a > 0\), as \(x
ightarrow\infty\), \(x^{2}\) becomes very large (positive) and \(ax^{2}\) dominates the other terms. So \(y=ax^{2}+bx + c
ightarrow\infty\) as \(x
ightarrow\infty\).

Answer:

• Absolute Max: None (or does not exist)
• Relative Max: None (or does not exist)
• Absolute Min: The \(y\) - value of the vertex (exists, as the parabola has a lowest point)
• Relative Min: The \(y\) - value of the vertex (same as absolute min for upward - opening parabola)
• As \(x

ightarrow-\infty\), \(f(x)
ightarrow\infty\)

• As \(x

ightarrow\infty\), \(f(x)
ightarrow\infty\)