Problem 94 - Precalculus : Building Concepts and Connections

Question

The range of a quadratic function \(g(x)=a x^{2}+b x+c\) is given by \((-\infty, 2] .\) Is \(a\) positive or negative? Justify your answer.

Step-by-Step Solution

Answer

The coefficient \(a\) of the quadratic function \(g(x) = ax^{2} + bx + c\) will be negative.

Step 1: Range interpretation

Firstly, we need to understand what the given range indicates. As the range is from -infinity to 2 included, it demonstrates that 2 is the maximum of the function and there's no upper limit to how low the function can go. Therefore, the parabola opens downwards.

Step 2: Relation to quadratic coefficient

In a quadratic function, the coefficient \(a\) of \(x^{2}\) determines the direction of the parabola. If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, the parabola opens downwards.

Step 3: Conclusion

Since the given function has a range that suggests an downward-opening parabola, the coefficient \(a\) of the quadratic function should be negative.

Key Concepts

Parabola Direction

Understanding the direction in which a parabola opens is crucial when studying quadratic functions. The parabola is the graph of a quadratic function expressed as \(g(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero.

When the coefficient \(a\) is positive, the parabola opens upwards, resembling a 'U' shape. This orientation indicates that the function has a minimum value, which is its lowest point on the graph. Conversely, if \(a\) is negative, the parabola opens downwards, similar to an upside-down 'U', and the function possesses a maximum value at its highest point.

In the case of the exercise where the range is specified as \((-\text{infinity},2]\), it is clear that the function reaches a maximum value and thus the parabola must open downwards. This insight directly leads us to understand that for the given function, the quadratic coefficient \(a\) must indeed be negative.

Quadratic Coefficient

The quadratic coefficient in the standard form of a quadratic equation, \(g(x) = ax^2 + bx + c\), is denoted by \(a\). Its value plays a decisive role in determining various attributes of the quadratic function, most notably the direction of the parabola as previously mentioned.

The sign of \(a\) influences not just the direction but also the width and the steepness of the parabola. A larger absolute value of \(a\) results in a narrower parabola, suggesting a steeper curve. On the other hand, a smaller absolute value causes the parabola to be wider and less steep.

In our exercise, analyzing the range, along with knowing the parabola opens downwards, definitively tells us that \(a\) is negative. This information not only indicates direction but also impacts the graph's shape and the function's behavior as it translates across the coordinate plane.

Range of a Function

The range of a function represents all the possible output values (\(y\)-values) a function can produce. For a quadratic function in the form \(g(x) = ax^2 + bx + c\), if the leading coefficient \(a\) is positive, the range is \([y_{min}, \text{infinity})\), with \(y_{min}\) being the minimum value attained by the function. Conversely, when \(a\) is negative, the range becomes \((-\text{infinity}, y_{max}]\), where \(y_{max}\) is the maximum value.

In the example exercise, the provided range \((-\text{infinity}, 2]\) tells us the highest value of \(g(x)\) that can be achieved is 2. This pivotal point where f(x) reaches its maximum is known as the vertex of the parabola in a downward-opening quadratic function. Therefore, by linking the range given to the nature of the function's graph, one can infer critical characteristics of the quadratic equation in question.