Problem 166 - Elementary Algebra

Question

In the following exercises, determine if the parabola opens up or down. $$ y=6 x^{2}+2 x+3 $$

Step-by-Step Solution

Answer

The parabola opens upwards.

Step 1: Identify the Quadratic Equation's Coefficients

The general form of a quadratic equation is given by \[ y = ax^2 + bx + c \]. The coefficients in the equation \[ y = 6x^2 + 2x + 3 \] are \( a = 6 \), \( b = 2 \), and \( c = 3 \).

Step 2: Determine the Sign of the Leading Coefficient

The direction in which the parabola opens is determined by the sign of the leading coefficient \( a \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), the parabola opens downwards. In this case, \( a = 6 \), which is positive.

Step 3: Conclusion

Since the coefficient \( a = 6 \) is positive, the parabola opens upwards.

Key Concepts

Parabola Direction

Understanding how a parabola opens is essential to mastering quadratic equations. Parabolas are the graphical representation of quadratic equations like \(y = ax^2 + bx + c\).
The direction of the parabola—whether it opens upwards or downwards—is determined by the sign of the 'a' coefficient in the quadratic equation.
When you have a quadratic equation:

• If \(a > 0\), the parabola opens upward.
• If \(a < 0\), the parabola opens downward.

In this specific exercise, \(a = 6\), which is a positive number. Thus, the parabola opens upwards. This directional property holds regardless of the values of \(b\) and \(c\).

Leading Coefficient

The leading coefficient is the coefficient of the \(x^2\) term in a quadratic equation, and it is denoted as 'a'. In our example, the quadratic equation is: \(y = 6x^2 + 2x + 3\).
Here, the leading coefficient is 6. The leading coefficient is crucial because it influences not only the direction but also the width and steepness of the parabola.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.
• The larger the absolute value of 'a', the steeper or narrower the parabola. Conversely, the smaller the absolute value of 'a', the wider the parabola.

For this exercise, since \(a = 6\) is positive and relatively large, the parabola will be steep and open upwards.

Upward Parabola

An upward parabola is a parabola that opens towards the sky, forming a 'U' shape. This occurs when the leading coefficient 'a' is positive.
In our example of \(y = 6x^2 + 2x + 3\), since \(a = 6\) is positive, the parabola is confirmed to open upwards.
Here are key characteristics of an upward parabola:

• The vertex (the lowest point of the parabola) is at the bottom of the 'U'.
• Both arms of the parabola extend upwards indefinitely.
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• The vertex represents the minimum value of the quadratic function.

The opening upwards ensures that for any point on this parabola, other than the vertex, the value of 'y' increases as you move away from the vertex. This property is vital for understanding the behavior of quadratic functions in various applications.