Problem 3 - Intermediate Algebra

Question

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$x=-y^{2}-y+2$$

Step-by-Step Solution

Answer

The parabola opens to the left.

Step 1: Determine the Form of the Equation

The given equation is \(x = -y^2 - y + 2\). This equation can be rearranged in the form \(x = ay^2 + by + c\). This indicates that the equation describes a parabola in terms of \(y\), suggesting that the parabola either opens to the left or right, unlike the typical \(y = ax^2 + bx + c\) form where it opens upward or downward.

Step 2: Identify Coefficient of the Quadratic Term

In the equation \(x = -y^2 - y + 2\), the coefficient of the \(y^2\) term is \(a = -1\). The sign of this coefficient determines the direction in which the parabola opens.

Step 3: Determine the Direction of Opening Based on Coefficient

Since the coefficient \(a = -1\) is negative, the parabola opens to the left. If \(a\) were positive, it would open to the right. In general, for parabolas in the form \(x = ay^2 + by + c\), a negative \(a\) value indicates an opening to the left.

Key Concepts

Parabola Equation

The core concept behind understanding a parabola's orientation starts with recognizing its equation form. Parabolas are represented in a mathematical equation known as the quadratic equation. Typically, for parabolas that open upwards or downwards, we encounter the standard form \(y = ax^2 + bx + c\). However, when a parabola opens to the left or right, the form of the equation switches to \(x = ay^2 + by + c\).

In the exercise given, \(x = -y^2 - y + 2\) is used to describe the parabola. Notice that \(y\) is squared, which automatically tells us that the parabola opens either to the left or the right instead of upwards or downwards. Identifying the way \(y\) or \(x\) is squared and where each variable is located is crucial when determining the orientation of the parabola. When the equation has the \(y^2\) term leading to an \(x =\) structure, the parabola's axis is horizontal.

Quadratic Coefficient

The quadratic coefficient in a parabola equation is the number situated in front of the squared term. This number is significant as it influences both the width and direction of the parabola.

In the equation \(x = -y^2 - y + 2\), the quadratic coefficient is \(a = -1\). The sign and magnitude of this coefficient affect the parabola's orientation.

• If \(a\) is a positive number, the parabola opens to the right (or upwards for \(y = ax^2 + bx + c\) forms).
• If \(a\) is a negative number, it opens to the left (or downwards in the typical form).
• The absolute value of \(a\) tells you how "stretched" or "squeezed" the parabola looks.

For example, if \(a\) had been \(-0.5\), the parabola would appear wider than if \(a\) were \(-1\). Understanding this coefficient is key to predicting the parabola's behavior without graphing it.

Direction of Opening

To determine the parabola's direction of opening, we look closely at the sign of the quadratic coefficient, \(a\). This guides us to whether the parabola opens leftward, rightward, upwards, or downwards. Remember that our given form is \(x = ay^2 + by + c\), which switches the standard vertical orientation.

In our specific example with \(a = -1\), the function \(x = -y^2 - y + 2\) tells us the parabola opens to the left because \(a\) is negative. If \(a\) had been positive, the equation would instead indicate that the parabola opens to the right.

• This consideration works the same in the equivalent vertical orientation \(y = ax^2 + bx + c\): positive \(a\) leads to upwards, negative \(a\) to downwards.
• Always cross-reference the given form to avoid misinterpreting the orientation.

With this understanding, one can visualize the parabola's direction just by inspecting the equation, bypassing the need for plotting or graphing every time.