Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-38: Find an equation for the indicated half of the parabola. $$ \text { Left half of }(x+3)^{2}=y+2 $$

Step-by-Step Solution

Answer

Equation: \(y = (x+3)^2 - 2\), for \(x \leq -3\).

Step 1: Identify the given equation

The given equation is \((x+3)^2 = y + 2\). This represents a parabola that opens upwards because the \(x\) term is squared and the \(y\) term is linear.

Step 2: Rearrange the equation

To understand the standard form better, rearrange the equation to solve for \(y\). Start by subtracting 2 from both sides to get \((x+3)^2 - 2 = y\). So the equation of the parabola is \(y = (x+3)^2 - 2\).

Step 3: Determine the vertex of the parabola

The parabola in vertex form is \(y = (x-h)^2 + k\), where \((h, k)\) is the vertex. Here \((h, k) = (-3, -2)\).

Step 4: Identify the direction and half of the parabola

Since we are asked for the left half of the parabola, it's important to note that the original parabola is symmetric with respect to the vertical line passing through \(x = -3\), and we only consider the values \(x \leq -3\).

Step 5: Write the equation for the left half

For the left half, we write the same equation but limit the domain of \(x\). Therefore, the equation for the left half of the parabola is \(y = (x+3)^2 - 2\) with \(x \leq -3\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a compelling way to express the equation of a parabola because it directly reveals the vertex of the parabola. This form is written as \( y = (x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
This specific equation format makes it easier to understand some key characteristics of the parabola, such as its direction and the position of its vertex.

• The term \((x-h)\) results in the parabola being horizontally shifted by \(h\) units. If \(h\) is positive, the shift is to the right; if negative, to the left.
• The \(k\) value causes the parabola to shift vertically. A positive \(k\) shifts it upwards, while a negative \(k\) shifts it downwards.

In our specific example, the equation \( y = (x+3)^2 - 2 \) is already in vertex form. This corresponds to a vertex at \((-3, -2)\). This vertex indicates the lowest point of the parabola since it opens upwards.

Domain Restriction

Domain restriction is a crucial concept when working with parabolas, especially when asked to find or analyze only a specific portion, such as the left half of a parabola.
In general, the domain of a parabola can be all real numbers, but restricting the domain allows us to focus on a portion of it.

• A domain restriction is often used when we want to consider only a specific segment or half of a symmetric function, like a parabola.
• This is done by specifying a range of \(x\) values for which the equation is valid.

For the exercise at hand, we are interested in the left half of the given parabola. This means we want the equation to represent only the values of \(x\) that are less than or equal to \(-3\), i.e., \( x \leq -3 \). Thus, the solution includes a restriction on the domain to align with the problem's requirement.

Symmetry of Parabolas

Symmetry in parabolas is a fascinating property that makes these shapes so unique and predictable. Parabolas are symmetrical about a vertical line that passes through their vertex.
This line of symmetry divides the parabola into two mirror-image halves.

• If the vertex form is \( y = (x-h)^2 + k \), then the line of symmetry is at \( x = h \).
• For our example equation \( y = (x+3)^2 - 2 \), the line of symmetry is \( x = -3 \).

Understanding the symmetry of parabolas helps us easily identify features of the parabola, like the maximum or minimum point, depending on its orientation. Symmetry is what helps us divide the parabola into its left or right half, which aligns perfectly with the exercise requirement to focus this time on the left half of the parabola. This means only the part of the graph where \( x \) is less than or equal to \(-3\) is considered.