Question

Can a quadratic function have a range of \((-\infty, \infty)\) ? Justify your answer.

Short answer

No, a quadratic function cannot have the range \((-\infty, \infty) \).

Step-by-step solution

Understand the Range of a Function

The range of a function is the set of all possible output values. For a quadratic function, which is generally written as \(f(x) = ax^2 + bx + c\), we need to determine if it can produce ALL possible output values from \(-\infty\) to \(+\infty\).

Analyze the Shape of Quadratic Functions

Quadratic functions create parabolas. The direction of a parabola (opening upwards or downwards) depends on the coefficient \(a\): if \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

Determine the Extent of the Range

For a parabola that opens upwards \((a > 0)\), the range is \[y \geq k\rightarrow [k, \infty)\]. For a parabola that opens downwards \((a < 0)\), the range is \[y \leq k\rightarrow (-\infty, k]\]. In both cases, the range does NOT cover \(-\infty\) to \(+\infty\).

Conclude Based on Analysis

Since neither direction of the parabola (\(a > 0\) or \(a < 0\)) covers all possible output values, a quadratic function CANNOT have the range \((-\infty, \infty) \).

Key concepts

Range of a Function

The range of a function is an important concept in understanding the behavior of mathematical functions. It represents all the possible output values (y-values) that a function can produce. For example, if you have a function that calculates the height of a ball over time, the range would be all the possible heights the ball can reach as time progresses.
For quadratic functions, represented by the formula \(f(x) = ax^2 + bx + c\), identifying the range involves considering the shape of the graph formed by this function, which is a parabola. This graph reveals the highest or lowest point that the function can reach, which helps us understand which y-values are possible.
To sum up, the range tells us a lot about how the function behaves and what values we can expect as outputs.

Parabola

An important shape in the study of quadratic functions is the parabola. The parabola is the graph of the quadratic function \(f(x) = ax^2 + bx + c\), and it has a distinct bowl-like shape. Depending on the coefficient \(a\), the parabola can open upwards or downwards:
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- When \(a > 0\), the parabola opens upwards, resembling a U-shape.
- When \(a < 0\), the parabola opens downwards, resembling an upside-down U-shape.
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The point at which the parabola changes direction is its vertex. For a parabola opening upwards, the vertex is the lowest point. For one opening downwards, the vertex is the highest point. The direction and the position of the vertex are crucial in understanding the function's range and behavior.
Parabolas are symmetrical around a vertical axis called the axis of symmetry, which goes through the vertex. Being familiar with these characteristics makes it easier to analyze and graph quadratic functions.

Coefficient Impact

The coefficients in a quadratic function significantly influence the shape and position of the parabola. Let's break down the impact of each coefficient (\(a\), \(b\), and \(c\)):
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- **Coefficient \(a\)**: This is the most influential coefficient. It determines the direction of the parabola (upwards if \(a > 0\) and downwards if \(a < 0\)). It also affects the 'width' of the parabola – a larger absolute value of \(a\) makes the parabola narrower, while a smaller absolute value makes it wider.
- **Coefficient \(b\)**: This coefficient influences the horizontal position of the vertex. Changing \(b\) shifts the vertex along the x-axis, but it doesn't change the direction or the width of the parabola.
- **Coefficient \(c\)**: This constant term moves the parabola up and down along the y-axis without altering its shape. Essentially, it affects the y-intercept of the graph – the point where the parabola crosses the y-axis.
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Understanding how each of these coefficients affects a parabola helps students manipulate and graph quadratic functions more easily. By adjusting these values, one can predict how the function's graph will look and where it will be positioned.