Question

If the graphof \(f(x)=a x^{2}+b x+c, a \neq 0\) has a maximum value at its vertex, which condition must be true? (a) \(-\frac{b}{2 a}>0\) (b) \(-\frac{b}{2 a}<0\) (c) \(a>0\) (d) \(a<0\)

Short answer

(d) a < 0

Step-by-step solution

Identify the Type of Parabola

The function given is a quadratic function in the form of a parabola. For the parabola to have a maximum value at its vertex, it must open downwards.

Determine the Condition for the Parabola

A parabola opens downwards when the coefficient of the quadratic term, which is 'a', is less than zero. In mathematical terms, this means that the condition is: a < 0.

Conclusion

Since 'a' must be less than zero for the function to have a maximum value at its vertex, the correct choice is: (d) a < 0

Key concepts

quadratic function

A quadratic function is a type of function that can be written in the form \(f(x) = ax^2 + bx + c\), where \(a \eq 0\). The highest exponent of the variable 'x' is 2, making it a second-degree polynomial. In this expression, 'a', 'b', and 'c' are constants, with 'a' determining if the parabola opens upwards or downwards.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

The shape of a quadratic function is commonly known as a parabola. This distinct U-shaped graph plays a crucial role in understanding various algebraic concepts.

parabola

A parabola is the graph of a quadratic function and has a symmetrical shape. Parabolas can either open upwards or downwards depending on the sign of the coefficient 'a'. If \(a > 0\), the parabola is open upwards, forming a 'U' shape. If \(a < 0\), it opens downwards, forming an inverted 'U' shape. The nature of the parabola's opening determines if it has a minimum or maximum value at the vertex. Understanding this can help easily identify the highest or lowest point of the function, known as its vertex.

maximum value

In the context of a quadratic function, the maximum value refers to the highest point on the graph. For a parabola that opens downwards (where \(a < 0\)), this occurs at the vertex. The vertex represents the peak or highest point of the graph. The maximum value of a quadratic function can be found by evaluating the function at its vertex. The clear identification of the parabola's direction helps determine if you are dealing with a maximum or minimum value.

vertex

The vertex of a parabola is the point where the function reaches its maximum or minimum value. For the quadratic function \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula: \(-\frac{b}{2a}\). This value is where the function changes direction. If the parabola opens upwards, the vertex represents the minimum point. If it opens downwards (requiring \(a < 0\)), the vertex is the maximum point. The y-coordinate of the vertex can be found by substituting the x-coordinate back into the function.

coefficient 'a'

The coefficient 'a' in the quadratic function \(f(x) = ax^2 + bx + c\) is crucial as it dictates the direction in which the parabola opens and the nature of its vertex. When analyzing quadratic functions, remember:

• If \(a > 0\), the parabola opens upwards, and the vertex represents the minimum value of the function.
• If \(a < 0\), the parabola opens downwards, and the vertex represents the maximum value of the function.

Thus, for the graph of \(f(x) = ax^2 + bx + c\) to have a maximum value at the vertex, the condition must be \(a < 0\), ensuring the parabola opens downwards.