Figure 2.18 shows the graph of the polynomial f(x) = ax² + bx + c. Find the nature of a, b and c.
Video Explanation
Watch the video explanation below:
Given
The graph of the quadratic polynomial
f(x) = ax² + bx + c
is shown in Fig. 2.18.
Observations from the Graph
- The parabola opens downwards.
- The graph cuts the x-axis at two distinct points.
- The graph cuts the y-axis below the x-axis.
- The vertex lies to the left of the y-axis.
Solution
1. Sign of a
Since the parabola opens downwards,
a < 0
2. Sign of c
The y-intercept of the graph is c.
Since the graph cuts the y-axis below the x-axis,
c < 0
3. Sign of b
The axis of symmetry of the parabola is:
x = −b / (2a)
Since the vertex lies to the left of the y-axis and a < 0,
b > 0
Final Answer
From the given graph:
- a < 0
- b > 0
- c < 0
Conclusion
Thus, from the graph of the polynomial f(x) = ax² + bx + c shown in Fig. 2.18, we conclude that a is negative, b is positive and c is negative.