Problem 34 - A Graphical Approach to Precalculus with Limits

Question

For \(f(x)=d x^{2}-\frac{1}{2} d x+k, d \neq 0,\) find the \(x\) -coordinate of the vertex.

Step-by-Step Solution

Answer

The \(x\)-coordinate of the vertex is \(\frac{1}{4}\).

Step 1: Identify the vertex formula for a parabola

The general equation of a quadratic function is given by \(f(x) = ax^2 + bx + c\). The vertex \(x\)-coordinate of this parabola is found using the formula \(x = -\frac{b}{2a}\).

Step 2: Match the given function to the general form

The given function is \(f(x) = dx^2 - \frac{1}{2}dx + k\). Here, \(a = d\), \(b = -\frac{1}{2}d\), and \(c = k\).

Step 3: Substitute values into the vertex formula

Substitute \(a = d\) and \(b = -\frac{1}{2}d\) into the vertex formula \(x = -\frac{b}{2a}\):\[ x = -\frac{-\frac{1}{2}d}{2d} \]

Step 4: Simplify the expression

Simplify the expression:\[-\frac{-\frac{1}{2}d}{2d} = \frac{\frac{1}{2}d}{2d} = \frac{1}{2} \times \frac{d}{2d} = \frac{1}{4} \]

Step 5: Conclude the solution

The \(x\)-coordinate of the vertex of the function \(f(x) = dx^2 - \frac{1}{2}dx + k\) is \(\frac{1}{4}\).

Key Concepts

Parabola

When you think of a parabola, imagine the smooth U-shaped curve you often see graphically representing quadratic functions. This curve is quite special as it symmetrically opens either upward or downward. A parabola holds a lot of information about the quadratic function it represents, such as the vertex, which is its turning point either at its maximum or minimum.

• A parabola can open upwards like a cup (")") or downwards like a cap (")").
• It is defined mathematically as the graph of a quadratic function.
• The vertex of the parabola is its highest or lowest point based on its opening direction.

Understanding the shape and orientation of a parabola is crucial in studying quadratic functions. Observe that the wider the parabola, the smaller the coefficient of the quadratic term is, and vice versa.

Quadratic Function

A quadratic function is a polynomial function of degree 2. It is typically written in the standard form as: \[ f(x) = ax^2 + bx + c \] where \( a, b, \) and \( c \) are constants, and \( a \) is not equal to zero since it defines the curvature of the parabola.

• The term \( ax^2 \) defines the shape and direction of the parabola.
• The term \( bx \) shifts the parabola along the x-axis.
• The constant \( c \) moves the parabola up or down along the y-axis.

Quadratic functions are pivotal in various applications ranging from physics to finance due to their symmetry and predictable pattern. They are most acknowledged for their role in defining the paths of projectiles or optimizing financial profit scenarios.

Vertex Formula

The vertex of a quadratic function is a critical aspect because it provides valuable information about the function's behavior. The vertex formula is used to find the x-coordinate of this point, crucial for graphing and analyzing the function. The formula is: \[ x = -\frac{b}{2a} \] This formula is derived from the process of completing the square or using the properties of symmetry of parabolas. Here, \( b \) and \( a \) are coefficients from the standard form of the quadratic function.

• The vertex formula helps find where the function reaches its peak (maximum) or valley (minimum).
• Using this formula helps in transforming the function into its vertex form for easier graphing.

Accurately finding the vertex using this formula is indispensable in both mathematical analysis and practical problem-solving.

x-coordinate

In mathematics, especially when working with parabolas and quadratic functions, finding the x-coordinate of the vertex is a routine yet vital step. This x-coordinate indicates the line of symmetry of the parabola and is essential for sketching the graph accurately.

• The x-coordinate directs us to the position along the horizontal axis where the vertex (the highest or lowest point of a parabola) is situated.
• In using the vertex formula \( x = -\frac{b}{2a} \), the x-coordinate equates to this calculation, guiding plots and analyses.
• It allows for precise identification of important graph features like intercepts and symmetry.

Mastering how to calculate the x-coordinate of the vertex empowers students in both understanding function behavior and implementing this knowledge in real-world scenarios.