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Magazine Chapter 3: Problem 34
For \(f(x)=d x^{2}-\frac{1}{2} d x+k, d \neq 0,\) find the \(x\) -coordinate of the vertex.
Short Answer
Expert verified
The \(x\)-coordinate of the vertex is \(\frac{1}{4}\).
Step by step solution
01
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 \(x = -\frac{b}{2a}\).
02
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 \(c = k\).
03
Substitute values into the vertex formula
Substitute \(a = d\) and \(b = -\frac{1}{2}d\) into the vertex formula \(x = -\frac{b}{2a}\):\[ x = -\frac{-\frac{1}{2}d}{2d} \]
04
Simplify the expression
Simplify the expression:\[-\frac{-\frac{1}{2}d}{2d} = \frac{\frac{1}{2}d}{2d} = \frac{1}{2} \times \frac{d}{2d} = \frac{1}{4} \]
05
Conclude the solution
The \(x\)-coordinate of the vertex of the function \(f(x) = dx^2 - \frac{1}{2}dx + k\) is \(\frac{1}{4}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
