Problem 8 - Algebra 2

Question

Write each equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola. $$ x=y^{2}+14 y+20 $$

Step-by-Step Solution

Answer

Vertex: (-29, -7); Axis of symmetry: y = -7; Opens to the right.

Step 1: Reorder the equation

Start with the given equation: \[ x = y^2 + 14y + 20 \]To convert this into the standard form of a parabola, which is \( (y - k)^2 = 4p(x - h) \), we need to complete the square for the \( y \) terms.

Step 2: Complete the square

Take the \( y^2 + 14y \) part of the equation. To complete the square, add and subtract \( (\frac{14}{2})^2 = 49 \) inside the equation:\[ x = (y^2 + 14y + 49) - 49 + 20 \]This becomes:\[ x = (y + 7)^2 - 29 \].

Step 3: Rearrange into standard form

Now, rearrange the equation to resemble the standard parabolic form \( (y - k)^2 = 4p(x - h) \):\[ (y + 7)^2 = x + 29 \]This means:\[ (y - (-7))^2 = 1(x - (-29)) \]Thus, the equation is now in the form \( (y - k)^2 = 4p(x - h) \).

Step 4: Identify the vertex

The standard form \( (y - k)^2 = 4p(x - h) \) shows the vertex at \((h, k)\). From the equation \( (y + 7)^2 = (x + 29) \), we identify:- Vertex \((h, k) = (-29, -7)\).

Step 5: Identify the axis of symmetry

The axis of symmetry for the parabola is the line \( y = k \), where \( k = -7 \). Thus, the axis of symmetry is- \( y = -7 \).

Step 6: Identify the direction of opening

Since the equation is \( (y - k)^2 = 4p(x - h) \) and there is a positive coefficient in front of \( x - h \), the parabola opens to the right. Therefore, the direction of opening is- Right.

Key Concepts

Standard Form

When working with parabolas, it's crucial to understand the standard form. The equation for a parabola in standard form is often written differently depending on its orientation on the coordinate plane. In the case of horizontal parabolas, such as \[(y - k)^2 = 4p(x - h),\] this configuration is what we achieved in the solution.The standard form reveals significant information:

• The expression \((y - k)^2\) indicates the parabola opens horizontally.
• The parameter \(h\) and \(k\) are derived from completed square form and reflect the parabola's translation from the origin.
• The number \(4p\) indicates the distance to the focus.

Simply rearranging can transform a seemingly complex equation into this manageable, interpretable form.

Vertex

The vertex of a parabola is a pivotal feature, as it represents the maximum or minimum point, depending on the parabola's orientation and the direction in which it opens. For the standard form \((y - k)^2 = 4p(x - h)\), the vertex is easy to identify at \((h, k)\).In the transformed equation \((y + 7)^2 = x + 29\), the vertex is found at:

• \(h = -29\)
• \(k = -7\)

Thus, the vertex point is \((-29, -7)\). This point acts as the dividing line where the parabola changes direction, offering symmetry around it.

Axis of Symmetry

The axis of symmetry is an imaginary line that mirrors each half of a parabola onto the other. For horizontal parabolas with equation form \((y - k)^2 = 4p(x - h)\), this line is always represented as \(y = k\). It's a vertical line intersecting the vertex.In our exercise, with a vertex of \((-29, -7)\), the axis of symmetry becomes:

• \(y = -7\)

This line is essential for understanding the parabola's structure, as each point on one side of this axis has an equidistant and corresponding point on the other.

Direction of Opening

Understanding the direction of opening is essential to graphically interpreting a parabola. For horizontal parabolas, the direction is determined by whether the \(4p\) term in \((y - k)^2 = 4p(x - h)\) is positive or negative.In our equation, \((y + 7)^2 = x + 29\), you can observe that \(4p = 1\), which is a positive value.

• When \(4p > 0\), the parabola opens to the right.
• When \(4p < 0\), the parabola opens to the left.

Therefore, since we have a positive \(4p\), this parabola's opening direction is to the right.