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Chapter 4: Problem 6

\(g(x)=\sqrt{3}-12 x+13 x^2\)

Short Answer

Expert verified
The roots and the vertex of the parabola formed by the given function are found by using the relevant formulas. The roots will give the x-values where the function intersects the x-axis and the vertex gives the maximum or minimum value of the function.

Step by step solution

01

Identify the Coefficients

Identify the coefficients from the quadratic equation. The standard quadratic equation is \(ax^2+bx+c\). In the given equation, \(a = 13\), \(b = -12\), and \(c = \sqrt{3}\).
02

Find the Roots

The roots of a quadratic equation are given by the formula: \(-b \pm \sqrt{b^2 - 4ac}/2a\). Substitute \(a = 13\), \(b = -12\), and \(c = \sqrt{3}\) into the formula to get the roots.
03

Find the Vertex

The vertex of a parabola represented by the equation \(ax^2 + bx + c\) is given by \(-b/2a, g(-b/2a)\). Substitute \(a = 13\) and \(b = -12\) into the formula to get the x-coordinate of the vertex. Then substitute this x-coordinate into the original function \(g(x)\) to get the y-coordinate.

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Key Concepts

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

Roots of Quadratic Equation
Understanding the roots of a quadratic equation is crucial. The roots, sometimes called solutions, are the values of \(x\) that satisfy the equation when the quadratic is set to zero. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the roots can be found using the quadratic formula:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
To apply this formula, we take these steps:
  • Identify the coefficients \(a\), \(b\), and \(c\).
  • Substitute these values into the quadratic formula.
  • Calculate the discriminant \(b^2 - 4ac\).
  • Determine whether the discriminant is positive, zero, or negative. - A positive discriminant means two real roots; zero indicates one real root; and negative means no real roots (only complex roots).
The roots tell us where the graph of the quadratic function intersects the x-axis. This is fundamental in graph analysis.
Vertex of a Parabola
The vertex of a parabola is a key feature and represents the peak or the lowest point of the graph, depending on the parabola's orientation. For the quadratic function \(g(x) = ax^2 + bx + c\), the vertex is found using this formula:
  • X-coordinate of the vertex: \(-b/(2a)\)
  • Y-coordinate: Substitute the x-coordinate back into the function \(g(x)\)
Here's how:
  • Calculate the x-coordinate by finding \(-b/(2a)\).
  • Plug this value into \(g(x)\) to find the y-coordinate.
The vertex (x, y) helps in sketching the parabola and understanding its shape. If \(a > 0\), the parabola opens upwards, making the vertex the lowest point. If \(a < 0\), it opens downwards, and the vertex is the highest point.This axis of symmetry around the vertex splits the parabola into two equal halves.
Coefficients of Quadratic Equation
Coefficients in a quadratic equation are the numbers beside the terms in the expression \(ax^2 + bx + c\). Each coefficient serves a specific purpose:
  • \(a\) is the leading coefficient that describes the parabola's direction and width. A larger absolute value of \(a\) makes the parabola narrower, while a smaller value widens it.
  • \(b\) affects the position of the vertex horizontally, moving it left or right along the x-axis.
  • \(c\) is the constant term that moves the graph vertically up or down and represents the y-intercept when \(x=0\).
Determining these coefficients is the first step in analyzing or graphing a quadratic equation. Each plays a role in shaping the parabola's appearance and its interaction with the x-axis.

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