Problem 54 - Algebra 2

Question

Write each equation in vertex form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=\frac{1}{2} x^{2}+6 x+18 $$

Step-by-Step Solution

Answer

Vertex: (-6, 0); Axis: x = -6; Opens upwards.

Step 1: Identify the standard form equation

The given quadratic equation is in the standard form: \[ y = \frac{1}{2}x^2 + 6x + 18 \] Our task is to rewrite this equation in the vertex form.

Step 2: Rewrite in vertex form

The vertex form of a quadratic equation is given by \[ y = a(x-h)^2+k \] where \((h, k)\) is the vertex of the parabola. To convert the given equation to vertex form, we will complete the square.

Step 3: Factor out the coefficient of \(x^2\)

Start by factoring out \(\frac{1}{2}\) from the quadratic and linear terms:\[ y = \frac{1}{2}(x^2 + 12x) + 18 \]

Step 4: Complete the square

To complete the square for the expression inside the parenthesis, take half of the coefficient of \(x\) (which is 12), square it, and add and subtract it inside the parenthesis.Half of 12 is 6, and 6 squared is 36. We have:\[ y = \frac{1}{2}(x^2 + 12x + 36 - 36) + 18 \]This simplifies to:\[ y = \frac{1}{2}((x+6)^2 - 36) + 18 \]

Step 5: Distribute and simplify

Distribute the \(\frac{1}{2}\) and simplify:\[ y = \frac{1}{2}(x+6)^2 - 18 + 18 \]\[ y = \frac{1}{2}(x+6)^2 \]This is the vertex form of the equation.

Step 6: Identify vertex, axis, and direction

From the vertex form \(y = \frac{1}{2}(x+6)^2\), we identify: - Vertex: \((-6, 0)\)- Axis of symmetry: \(x = -6\)- Direction of opening: Upwards (since the coefficient of \( (x+6)^2 \) is positive).

Key Concepts

Vertex Form

When dealing with quadratic equations, the vertex form is especially useful. It looks like this: \( y = a(x-h)^2 + k \). Here, \((h, k)\) represents the vertex of the parabola, giving us a clear picture of its highest or lowest point. In the vertex form, the value \(a\) still dictates the width and the direction of opening of the parabola, just like in the standard form. Transforming a quadratic equation from standard form to vertex form helps in easily identifying these characteristics.

• The vertex \((h, k)\) shows exactly where the parabola makes its sharpest turn.
• Rewrite equations into vertex form primarily through completing the square.

Converting an equation into vertex form is an approachable and systematic process that enhances our understanding of the parabola's graphical representation.

Completing the Square

The process of completing the square transforms a quadratic equation into vertex form. This technique involves several key steps:

• First, factor out any coefficient of \(x^2\) from the quadratic and linear terms if needed.
• Next, focus on the coefficient of \(x\).
• Take half of this coefficient, square it, and then add and subtract this square within the equation.

By restructuring the equation this way, you create a perfect square trinomial within the parentheses. For example, with the quadratic \(y = \frac{1}{2}x^2 + 6x + 18\), you factor \(\frac{1}{2}\), then manipulate the terms inside to complete the square. This manipulation crafts the form needed to identify the vertex and other characteristics of the parabola.

Parabola Characteristics

Parabolas, which are represented by quadratic equations, have distinctive features based on their vertex form \(y = a(x-h)^2+k\). Here’s what each term represents and implies:

• The vertex \((h, k)\) provides the "peak" or "valley" of the parabola, depending on the direction of the opening.
• The direction of opening is influenced by the sign and magnitude of \(a\); a positive \(a\) ensures an upwards opening, while a negative \(a\) leads to a downwards opening.
• The magnitude of \(a\) affects the width of the parabola—larger values make it narrower, while smaller values make it wider.

The vertex form offers a simple way of extracting these features, allowing students to graphically visualize and understand the overall layout and behavior of a parabola just by inspecting the equation.

Axis of Symmetry

A key characteristic of parabolas is their axis of symmetry. This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. The axis of symmetry for any parabola in the vertex form \(y = a(x-h)^2+k\) is simply \(x = h\). This indicates that the vertex \((h, k)\) is perfectly centered along this axis. For the equation \(y = \frac{1}{2}(x+6)^2\), the axis is \(x = -6\), showing the parabola is symmetrically aligned around this line.

Recognizing the axis of symmetry aids in plotting points on the parabola, making graph drawing more accurate and intuitive. For any point \(x_1=y_1\) on one side of the axis, \(x_2 = 2h-x_1\) will yield the same \(y\) value on the opposite side.