Question

The equation above represents the graph of a parabola in the \(x y\)-plane. Which of the following represents an equivalent form of the equation that includes the minimum value of \(y\) as a constant? A) \(y-28=x(x+16)\) B) \(y=x^2+2(8 x+14)\) C) \(y=x(x+16)+28\) D) \(y=(x+8)^2-36\)

Short answer

The correct option is D) \(y = (x + 8)^2 - 36\), since it represents the equation in vertex form, including the minimum value of \(y\) as a constant.

Step-by-step solution

\(y = x^2 + 16x + 28\) #Step 2: Rewrite the equation in vertex form# To rewrite the equation in vertex form, we need to complete the square. The vertex form of a quadratic is \(y = a(x-h)^2+k\).

Find the value to complete the square

To complete the square, we need to find the value that can be added and subtracted inside the square(parentheses). We will take half of the coefficient of x and square it: \(\frac{1}{2}(16)=8; \;8^2=64\)

Complete the square in the given equation

Add and subtract the value found in the previous step (64) in the equation: \(y = x^2 + 16x + 64 - 64 + 28\) Combine constant terms: \(y = x^2 + 16x + 64 - 36\) Now the equation looks like this: \(y = (x^2 + 16x + 64) - 36\)

Write the equation in vertex form

The equation in vertex form becomes: \(y = (x + 8)^2 - 36\) Comparing this form with the given options, we find that it matches option D. #Step 3: Choose the correct option# The correct option is:

D) \(y = (x + 8)^2 - 36\)

Key concepts

Vertex Form

The vertex form of a quadratic equation is crucial when dealing with parabolas. It appears as: \[ y = a(x-h)^2+k \]where:

• \(a\) determines the width and the direction of the parabola (upward for \(a>0\) and downward for \(a<0\)).
• \(h\) and \(k\) indicate the vertex of the parabola, which is the highest or lowest point depending on its orientation.

This form is incredibly useful because it directly tells you the vertex \((h, k)\). If you're trying to find how a parabola shifts or its minimum or maximum position, vertex form offers a tidy solution. The transformation from standard form to vertex form often involves completing the square, giving a new perspective on how the graph presents itself.
When determining which form of the equation illuminates the vertex, vertex form stands out as it directly showcases the values \(h\) and \(k\). These values show where the parabola reaches its extremes, allowing for easy analysis of the graph's behavior.

Completing the Square

Completing the square is a technique used to change a quadratic expression into perfect square form. This is key to rewriting the quadratic equation into vertex form.
This technique involves:

• Taking the coefficient of the \(x\)-term, dividing it by 2, and then squaring it.
• Adding and subtracting this new square within the equation.

For example, with the equation \(x^2 + 16x\), take \(\frac{16}{2} = 8\) and square it to get 64. By adding and subtracting this inside the equation, it transforms into a perfect square, i.e., \((x+8)^2\).
Completing the square not only aids in finding the vertex form but also clarifies the symmetry and axis of symmetry of the parabola. This technique fundamentally involves manipulating the equation to reveal its geometric properties through algebraic transformation, allowing us to grasp how the equation maps onto the graph.

Quadratic Equation

A quadratic equation is a second-degree polynomial, typically represented as \(ax^2 + bx + c = 0\). This is a foundational concept in algebra and its graph is known as a parabola.
The general properties include:

• It opens upwards if \(a > 0\) and downwards if \(a < 0\).
• The highest or lowest point of this graph, the vertex, provides insight into the parabola's minimum or maximum value.
• The solutions to the quadratic equation can be found using methods like factoring, completing the square, or the quadratic formula.

Each form of a quadratic equation highlights different aspects of its graph. While the standard form \(ax^2 + bx + c\) is straightforward for operations like addition or multiplication, the vertex form is excellent for graphing and understanding transformations. Recognizing these equations and their properties is vital for solving and interpreting various mathematical and physical problems.

Minimum Value

In the context of a parabola, its minimum value is the lowest y-coordinate on the graph, occurring at the vertex if the parabola opens upwards.
When expressed in vertex form \(y = a(x-h)^2+k\), the minimum or maximum value is readily identifiable:

• If \(a>0\), the minimum value is \(k\).
• If \(a<0\), the parabola opens downward and instead possesses a maximum.

Identifying whether a parabola has a minimum or maximum is determined exclusively by the sign of \(a\). For the exercise in question, the transformation into vertex form demonstrated that the minimum value of the parabola occurs at \(k = -36\). This insight directly answers the exercise query by revealing the smallest y-coordinate the parabola reaches. Understanding where these minimums and maximums occur aids in tackling broader questions involving optimization and real-world applications of quadratic models.