Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-12: Express \(f(x)\) in the form \(a(x-h)^{2}+k\). $$ f(x)=-x^{2}-4 x-8 $$

Step-by-Step Solution

Answer

The function in vertex form is \( f(x) = -(x + 2)^2 - 4 \).

Step 1: Identify the Form

The given function is a quadratic in standard form: \[ f(x) = -x^2 - 4x - 8 \] To express it in the vertex form \( f(x) = a(x-h)^2 + k \), we need to complete the square.

Step 2: Factor the Leading Coefficient

First, factor out the leading coefficient (-1) from the quadratic and linear terms:\[ f(x) = -(x^2 + 4x) - 8 \]

Step 3: Complete the Square

To complete the square for \( x^2 + 4x \), take half of the coefficient of \( x \), square it, and add then subtract it inside the bracket:Half of \( 4 \) is 2, and \( 2^2 = 4 \).Add and subtract 4 inside the bracket:\[ f(x) = -(x^2 + 4x + 4 - 4) - 8 \]

Step 4: Simplify the Expression

Simplify within the bracket and factor the perfect square trinomial:\[ f(x) = -((x + 2)^2 - 4) - 8 \] Distribute the -1 into the completed square:\[ f(x) = -(x + 2)^2 + 4 - 8 \] Combine like terms:\[ f(x) = -(x + 2)^2 - 4 \]

Step 5: Identify the Vertex Form

Now the expression is in vertex form:\( f(x) = -(x + 2)^2 - 4 \)This means \( a = -1 \), \( h = -2 \), and \( k = -4 \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is expressed as \( f(x) = a(x-h)^2 + k \). This form is particularly useful in revealing the properties of the quadratic function. The letters \( h \) and \( k \) in the equation represent the coordinates of the vertex of the parabola. The value of \( a \) determines the direction and the width of the parabola.

• The vertex \((h, k)\) tells us the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
• If \( a \) is positive, the parabola opens upward, making the vertex the minimum point.
• If \( a \) is negative, as in this example with \( a = -1 \), the parabola opens downward, and the vertex is the maximum point.

Understanding the vertex form helps in graphing the function and analyzing its features quickly. In this exercise, by converting the quadratic into vertex form, we can easily identify its vertex at \((-2, -4)\), indicating how the parabola is positioned on the graph.

Completing the Square

Completing the square is a technique used to convert a quadratic equation from its standard form into its vertex form. This method is essential when you want to manipulate the equation to make graphing simpler or to solve for variables in certain contexts.

To complete the square:

• First, factor out the leading coefficient if it's not 1. In our example, the leading coefficient is -1, so we factor it out of the first two terms.
• Next, take the coefficient of the linear term, divide it by 2, and square the result. This value helps form a perfect square trinomial.
• In our example, the coefficient of \( x \) is 4. Dividing it by 2 gives 2, and squaring it results in 4.
• Add and subtract this square inside the expression to maintain equality. Use this to form a perfect square trinomial, simplifying the bracketed term to \((x+2)^2\).

Completing the square transforms the quadratic into a form that highlights its vertex, making the graphing and interpretation of the function more manageable.

Standard Form

The standard form of a quadratic function is given by \( f(x) = ax^2 + bx + c \). This is the form that most quadratic equations are initially presented in. It is straightforward and easy for performing basic arithmetic operations, but it doesn't immediately provide information about the function's graph, such as the vertex or direction.

• The coefficients \( a \), \( b \), and \( c \) influence the shape and position of the parabola.
• If \( a \) is positive, the parabola opens upward; if negative, it opens downward.
• The vertex can be calculated from this form using formulas, but it's not as visually apparent as in the vertex form.

The standard form is essential for understanding the basic structure and broad behavior of a quadratic, but transforming to vertex form through techniques like completing the square can provide deeper insights.

Quadratic Equation Transformation

Quadratic equation transformation involves changing the form of a quadratic function to better understand or visualize its characteristics. This transformation usually refers to shifting from the standard form to other forms like vertex form or factored form.

• Completing the square is one method used to transform the equation into vertex form, which makes the vertex and symmetry of the parabola evident.
• Transforming into vertex form simplifies the process of graphing and solving quadratics.
• Changing the form can also reveal roots or solutions to a function clearly, depending on the need of the problem you are dealing with.

Such transformations make handling quadratic functions more dynamic and adaptable for various mathematical applications, enhancing our problem-solving abilities effectively.