Interpretation of the Graph of a Quadratic Polynomial

Given

The graph shown in Fig. 2.18 represents the quadratic polynomial

\[ f(x) = ax^2 + bx + c \]

The coordinates of the vertex are shown as

\[ \left(-\frac{b}{2a},\; -\frac{D}{4a}\right) \]

where \(D = b^2 – 4ac\) is the discriminant.

Solution

Step 1: Sign of \(a\)

From the graph, the parabola opens upwards.

Therefore,

\[ a > 0 \]

Step 2: Nature of the Discriminant

The vertex of the parabola lies below the x-axis.

So,

\[ -\frac{D}{4a} < 0 \]

Since \(a > 0\), the denominator \(4a\) is positive.

Hence, the numerator must be negative:

\[ -D < 0 \Rightarrow D > 0 \]

Therefore, the quadratic polynomial has two distinct real zeroes.

Step 3: Sign of \(c\)

From the graph, the curve cuts the y-axis above the x-axis.

So,

\[ f(0) = c > 0 \]

Conclusion

From the given graph of the quadratic polynomial,

\[ \boxed{a > 0,\quad D > 0,\quad c > 0} \]