Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-12: Express \(f(x)\) in the form \(a(x-h)^{2}+k\). $$ f(x)=-4 x^{2}+16 x-13 $$

Step-by-Step Solution

Answer

\(f(x) = -4(x-2)^2 + 3\)

Step 1: Identify the components of the quadratic

The given quadratic function is \(f(x) = -4x^2 + 16x - 13\). This function is in the form \(ax^2 + bx + c\), where \(a = -4\), \(b = 16\), and \(c = -13\).

Step 2: Complete the square

To express \(f(x)\) in the form \(a(x-h)^2 + k\), we need to complete the square. First, factor out the leading coefficient from the first two terms:\[f(x) = -4(x^2 - 4x) - 13\]Take half of the coefficient of \(x\) inside the parentheses, square it, and add and subtract it:The coefficient of \(x\) is -4, half of -4 is -2, and (-2)^2 is 4.\[f(x) = -4(x^2 - 4x + 4 - 4) - 13\]\[f(x) = -4((x - 2)^2 - 4) - 13\]

Step 3: Simplify the expression

Distribute the \(-4\) across the terms in the parentheses:\[f(x) = -4(x - 2)^2 + 16 - 13\]Combine the constant terms:\[f(x) = -4(x - 2)^2 + 3\]

Step 4: Write the final expression

The given quadratic function \(f(x) = -4x^2 + 16x - 13\) can be expressed in vertex form as:\[f(x) = -4(x-2)^2 + 3\]

Key Concepts

Completing the Square

Completing the square is a helpful method for transforming a quadratic function from its standard form, \(ax^2 + bx + c\), to a form called the vertex form, \(a(x-h)^2+k\).
This approach makes it easier to determine key properties about the quadratic graph, such as its vertex.

• First, always start by isolating the quadratic and linear terms. For a function such as \(f(x) = -4x^2 + 16x - 13\), factor out the coefficient of the \(x^2\) term from the first two terms: \(-4(x^2 - 4x) - 13\).


• Next, to "complete the square," take half of the coefficient of the "x" term. In this exercise, half of 4 is 2. Now, square this number to get 4. This transforms the expression into: \( -4((x^2 - 4x + 4) - 4) - 13\).


• Be sure to add and subtract the same number to maintain equality. By taking care to balance the equation, you can simplify it properly.

Completing the square is an essential step in changing a quadratic into a more useful form for analysis.

Vertex Form

The vertex form of a quadratic function is particularly useful as it gives direct insight into the vertex of the graph of the quadratic. The form is given by \(a(x-h)^2 + k\), where:

• \(h\) is the horizontal shift, or x-coordinate of the vertex.


• \(k\) is the vertical shift, or y-coordinate of the vertex.


• \(a\) determines the direction and width of the parabola.

For the function \(f(x) = -4(x-2)^2 + 3\):

• The vertex of the parabola is at point \((h, k) = (2, 3)\).


• The negative \(a\) value (-4) indicates that the parabola opens downwards, being vertically compressed.

Understanding the vertex form is crucial for sketching the graph of a quadratic function and determining its minimum or maximum points.

Polynomial Expressions

Quadratic functions are a type of polynomial expression, typically composed of terms involving variables raised to a power, constants, and coefficients, such as \(ax^2 + bx + c\).
Polynomial expressions feature various degrees based on the highest power of the variable, and quadratics are degree two.

• In general, polynomial expressions like the one given, \(-4x^2 + 16x - 13\), are commonly found in many scientific fields, including physics, and economics, due to their versatile nature.


• The coefficients \(-4\), \(16\), and the constant \(-13\) determine the graph's shape, direction, and other characteristics.

Grasping polynomial expressions is foundational to further mathematical studies and real-world applications. Recognizing that converting a quadratic into multiple forms, such as vertex form, is necessary for enhancing problem-solving efficiency.