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Chapter 3: Problem 4

Fill in the blank. For \(f(x)=a x^{2}+b x+c(a \neq 0),\) the \(x\) -coordinate of the ________ is \(-b /(2 a)\)

Short Answer

Expert verified
The x-coordinate of the vertex is -\(\frac{b}{2a}\).

Step by step solution

01

Identify the form of the quadratic function

Given the quadratic function in the form of y = f(x) = ax^2 + bx + c, it's clear that we are dealing with a standard quadratic equation.
02

Determine the vertex form

The vertex form of a quadratic function is useful for identifying key features of the parabola represented by the function. One way to find the vertex is by completing the square, but a more straightforward approach is needed here.
03

Use the formula for the vertex

The x-coordinate of the vertex of a quadratic function in standard form can be found using the formula: \(x = -\frac{b}{2a}\)
04

Substitute into the vertex formula

Directly substitute the given constants from the function \(f(x) = ax^2 + bx + c\) into the formula to find the x-coordinate of the vertex.
05

Conclusion

The x-coordinate of the vertex of the function \(f(x) = ax^2 + bx + c\) is explicitly given as \(-\frac{b}{2a}\).

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Key Concepts

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

vertex formula
The vertex of a parabola is a significant point that represents the maximum or minimum value of the quadratic function. This makes it crucial for graphing and understanding the behavior of the quadratic equation. The x-coordinate of the vertex can be found using the vertex formula: \(\[- \frac{b}{2a} \]\). Here, \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(f(x) = ax^2 + bx + c\).
  • \(a\) determines the parabola's direction and width.
  • \(b\) affects the vertex's location along the x-axis.
  • \(c\) is the y-intercept of the parabola.
Using the vertex formula allows you to efficiently locate the x-coordinate of the vertex, which is essential for plotting and analyzing the quadratic function.
quadratic equation
A quadratic equation is a second-order polynomial represented in the form \(ax^2 + bx + c = 0\). This is one of the most common forms of equations encountered in algebra. Key features of quadratic equations include:
  • It forms a parabola when graphed on a coordinate plane.
  • The coefficients \(a\), \(b\), and \(c\) determine the shape and position of the parabola.
  • Solutions to quadratic equations can be found using methods like factoring, completing the square, and the quadratic formula.
Understanding the structure and properties of quadratic equations is vital for solving them and interpreting their graphs.
parabola
A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function. Key characteristics of a parabola include:
  • It opens upward if \(a > 0\) and downward if \(a < 0\).
  • The vertex is the highest or lowest point on the parabola, depending on the direction it opens.
  • The axis of symmetry passes through the vertex and divides the parabola into two mirror-image halves.
  • The y-intercept is where the parabola crosses the y-axis, corresponding to the value of \(c\) in the quadratic equation.
Parabolas are fundamental in various fields, including physics, engineering, and economics, making a deep understanding of their properties highly beneficial.

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