Interpretation of the Graph of a Quadratic Polynomial
Given
The graph shown in Fig. 2.17 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 downwards.
Therefore,
\[ a < 0 \]
Step 2: Nature of the Discriminant
The vertex of the parabola lies above the x-axis.
So,
\[ -\frac{D}{4a} > 0 \]
Since \(a < 0\), the denominator \(4a\) is negative.
For the fraction to be positive, the numerator must also be negative:
\[ -D < 0 \Rightarrow D > 0 \]
Hence, the quadratic polynomial has two distinct real zeroes.
Step 3: Sign of \(c\)
The graph cuts the y-axis below the x-axis.
Therefore,
\[ f(0) = c < 0 \]
Conclusion
From the given graph of the quadratic polynomial,
\[ \boxed{a < 0,\quad D > 0,\quad c < 0} \]