1. Let f(x)=ax^2+bx+c be a quadratic function where a≠0 .Which of the following statements is true about the relationship between the vertex of f(x) and the value of a?
A. The vertex of f(x) is a minimum when a>0 . The vertex of is a maximum when a<0 .
B. The vertex of f(x) is a maximum when a>0 . The vertex of is a minimum when a<0 .
C. The vertex of f(x) is a minimum when a>0 or a<0
D. The vertex of f(x) is a maximum when a>0 or a<0
B. The vertex of f(x) is a maximum when a>0 . The vertex of is a minimum when a<0 .
The correct statement about the relationship between the vertex of f(x) and the value of a is:
B. The vertex of f(x) is a maximum when a > 0. The vertex of f(x) is a minimum when a < 0.
To determine the relationship between the vertex of the quadratic function f(x) and the value of a, we need to understand how the sign of a affects the shape of the parabola.
The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.
The vertex of a quadratic function is given by the formula (-b/2a, f(-b/2a)). In other words, the x-coordinate of the vertex is -b/2a, and the y-coordinate is f(-b/2a).
Now, let's consider the impact of the coefficient a on the shape of the parabola:
1. If a > 0:
- The quadratic term (ax^2) is positive, which means the parabola opens upwards.
- Since the parabola opens upwards, the vertex will be the lowest point on the graph, and hence it will be the minimum point.
- Therefore, the vertex of f(x) is a minimum when a > 0.
2. If a < 0:
- The quadratic term (ax^2) is negative, which means the parabola opens downwards.
- Since the parabola opens downwards, the vertex will be the highest point on the graph, and hence it will be the maximum point.
- Therefore, the vertex of f(x) is a maximum when a < 0.
Based on this analysis, the correct statement is:
A. The vertex of f(x) is a minimum when a > 0. The vertex of f(x) is a maximum when a < 0.
So, the answer is A.