<p>The graph of a quadratic polynomial <span class="math">y = ax^2 + bx + c; a, b, c \in \mathbb{R}</span> is as shown. Which one of the following is not correct?</p>
<p>(a) <span class="math">b^2 - 4ac > 0</span></p>
<p>(b) <span class="math">\frac{c}{a} > 0</span></p>
<p>(c) c is negative</p>
<p>(d) Abscissa corresponding to the vertex is <span class="math">\left(-\frac{b}{2a}\right)</span></p>
Step-by-Step Solution
Key Concept: From the graph, we can determine the sign of coefficients a, b, c by observing the parabola's orientation, y-intercept, vertex position, and number of real roots. The condition b² - 4ac > 0 indicates two distinct real roots, which must be verified against the visual representation.
<p><strong>Step 1: Determine the sign of a.</strong></p><p>From the graph, the parabola opens upward, so <strong>a > 0</strong>.</p><p><strong>Step 2: Determine the sign of c.</strong></p><p>The y-intercept occurs at y = c (when x = 0). From the graph, the parabola intersects the y-axis <strong>above the origin</strong>, so <strong>c > 0</strong>.</p><p><strong>Step 3: Analyze option (a): b² - 4ac > 0.</strong></p><p>The graph shows the parabola intersects the x-axis at <strong>two distinct points</strong>. This means the quadratic has two distinct real roots, which requires the discriminant to be positive: <strong>b² - 4ac > 0 ✓ CORRECT</strong></p><p><strong>Step 4: Analyze option (b): c/a > 0.</strong></p><p>Since a > 0 and c > 0, we have c/a > 0 <strong>✓ CORRECT</strong></p><p><strong>Step 5: Analyze option (c): c is negative.</strong></p><p>From Step 2, we established that c > 0 (the y-intercept is above the origin). Therefore, the statement "c is negative" is <strong>✗ INCORRECT</strong></p><p><strong>Step 6: Analyze option (d): Abscissa of vertex is -b/(2a).</strong></p><p>This is the standard formula for the x-coordinate of the vertex of any quadratic y = ax² + bx + c, regardless of the specific graph. <strong>✓ CORRECT</strong></p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C