If the diagram in Fig. 2.17 shows the graph of the polynomial f(x) = ax² + bx + c, then find the nature of a, b and c
Video Explanation
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Given
The graph of the quadratic polynomial:
f(x) = ax² + bx + c
is shown in Fig. 2.17.
Observation from the Graph
- The parabola opens upwards.
- The graph cuts the x-axis at two distinct points.
- The graph cuts the y-axis above the x-axis.
- The vertex lies to the right of the y-axis.
Solution
1. Sign of a
Since the parabola opens upwards,
a > 0
2. Sign of c
The y-intercept of the graph is c.
Since the graph cuts the y-axis above the x-axis,
c > 0
3. Sign of b
The axis of symmetry of the parabola is given by:
x = −b / (2a)
Since the vertex lies to the right 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. 2.17, we conclude that a is positive, b is negative and c is positive.