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Chapter 10: Problem 14

Consider the function \(f(x)=2 x^{2}+4 x-8\) a. What are \(a, b,\) and \(c ?\) b. Find \(-\frac{b}{2 a}\)

Short Answer

Expert verified
a = 2, b = 4, c = -8; \(-\frac{b}{2a} = -1\).

Step by step solution

01

Identify Coefficients

The given function is a quadratic function in the standard form: \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are the coefficients. From the function \( f(x)=2x^2+4x-8 \), we can identify:- \( a = 2 \)- \( b = 4 \)- \( c = -8 \).
02

Calculate \(-\frac{b}{2a}\)

We need to find the value of \(-\frac{b}{2a}\). Using the identified coefficients where \( b = 4 \) and \( a = 2 \), substitute them into the expression:\[-\frac{b}{2a} = -\frac{4}{2 \times 2} = -\frac{4}{4} = -1\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficients of Quadratic Equations
When dealing with quadratic functions, understanding the coefficients is crucial. A quadratic function is typically represented in its algebraic form:
  • \( f(x) = ax^2 + bx + c \)
Here, the letters \( a \), \( b \), and \( c \) stand as the coefficients.
These coefficients provide essential information:
  • \( a \): the coefficient of \( x^2 \) determines the parabola's vertical stretch or compression and indicates the direction it opens.
  • \( b \): the coefficient of \( x \) influences the linear component of the function.
  • \( c \): the constant term shifts the graph vertically.
In practice, these coefficients are identified by comparing the given quadratic with its standard form.
For example, in the function \( f(x)=2x^2+4x-8 \), the coefficients are \( a = 2 \), \( b = 4 \), and \( c = -8 \). Recognizing these values allows a deeper understanding of the parabola's behavior.
Vertex of a Parabola
The vertex of a parabola is a key feature of a quadratic equation. It's the point where the parabola changes direction, which can be identified using the formula:
  • \( -\frac{b}{2a} \)
This formula helps find the x-coordinate of the vertex. For the function \(f(x)=2x^2+4x-8\), the coefficients \(a = 2\) and \(b = 4\) are used:
  • \( -\frac{b}{2a} = -\frac{4}{2 \times 2} = -1 \)
This calculation determines that the x-coordinate of the vertex is \(-1\). Once the x-coordinate is known, substitute back into the function to find the y-coordinate.
Understanding the vertex allows you to know the highest or lowest point on the graph and provides insight into the symmetry and position of the parabola.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is the foundation upon which many other calculations are built. It is written as:
  • \( f(x) = ax^2 + bx + c \)
This form offers a clear framework for identifying the nature and characteristics of the parabola.
Having it in this format is vital for several reasons:
  • It simplifies the process of identifying coefficients \( a \), \( b \), and \( c \).
  • It facilitates the calculation of the vertex, axis of symmetry, and even the parabola's roots, if factored or solved further.
  • It provides a straightforward method to assess how changes in coefficients affect the graph's shape and position.
In essence, converting a quadratic equation into its standard form is a step towards unlocking a deeper understanding of its graphical properties and real-world applications.

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