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Magazine Chapter 2: Problem 46
Find the roots and sketch the graphs of the quadratic functions:\(3 x^{2}-3 x-1\)
Short Answer
Expert verified
Roots are \(x = \frac{3 + \sqrt{21}}{6}\) and \(x = \frac{3 - \sqrt{21}}{6}\); graph is an upward-opening parabola.
Step by step solution
01
Identify the Quadratic Equation Format
The given equation is in the standard quadratic form \(ax^2 + bx + c = 0\). In the equation \(3x^2 - 3x - 1 = 0\), we have \(a = 3\), \(b = -3\), and \(c = -1\).
02
Calculate the Discriminant
To find the roots, first calculate the discriminant \(\Delta = b^2 - 4ac\). Substituting the values, we get \((-3)^2 - 4 \cdot 3 \cdot (-1) = 9 + 12 = 21\). Since the discriminant is positive, there will be two distinct real roots.
03
Find the Roots Using the Quadratic Formula
The quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) provides the roots. Substituting the known quantities, \(x = \frac{-(-3) \pm \sqrt{21}}{2 \cdot 3}\) simplifies to \(x = \frac{3 \pm \sqrt{21}}{6}\). Therefore, the roots are \(x = \frac{3 + \sqrt{21}}{6}\) and \(x = \frac{3 - \sqrt{21}}{6}\).
04
Determine the Vertex of the Parabola
The vertex \((h, k)\) of a quadratic function \(ax^2 + bx + c\) is given by \(h = -\frac{b}{2a}\). For this quadratic, \(h = \frac{-(-3)}{2 \cdot 3} = \frac{3}{6} = \frac{1}{2}\). Substitute this back into the original equation to find \(k: k = 3\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) - 1 = \frac{3}{4} - \frac{3}{2} - 1 = -\frac{7}{4}\). Thus, the vertex is \(\left(\frac{1}{2}, -\frac{7}{4}\right)\).
05
Sketch the Graph
The graph of a quadratic function is a parabola. Since \(a = 3 > 0\), the parabola opens upwards. Plot the vertex \(\left(\frac{1}{2}, -\frac{7}{4}\right)\), and indicate the roots on the x-axis where the parabola crosses the axis: \(x = \frac{3 + \sqrt{21}}{6}\) and \(x = \frac{3 - \sqrt{21}}{6}\). Connect these points to sketch the parabolic shape.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discriminant
In quadratic equations, the discriminant plays a crucial role in determining the nature of the roots. The discriminant is given by the formula \( \Delta = b^2 - 4ac \). It serves as an indicator of how the parabola intersects the x-axis, if at all.
For the quadratic equation \(3x^2 - 3x - 1 = 0\), the discriminant is calculated as follows:
This positive value implies two distinct real roots exist, indicating the parabola will intersect the x-axis at two points.
For the quadratic equation \(3x^2 - 3x - 1 = 0\), the discriminant is calculated as follows:
- \(a = 3\)
- \(b = -3\)
- \(c = -1\)
This positive value implies two distinct real roots exist, indicating the parabola will intersect the x-axis at two points.
Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is given by \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \), where \(\Delta\) is the discriminant. This formula provides a straightforward method to solve any quadratic equation.
For our example \(3x^2 - 3x - 1 = 0\), we substitute the values:
For our example \(3x^2 - 3x - 1 = 0\), we substitute the values:
- \(b = -3\)
- \(\Delta = 21\)
- \(a = 3\)
- \(x = \frac{3 + \sqrt{21}}{6}\)
- \(x = \frac{3 - \sqrt{21}}{6}\)
Parabola
Graphically, every quadratic equation corresponds to a parabola. A parabola is a symmetric curve and the shape of its graph depends on the coefficient \(a\) in the equation \(ax^2 + bx + c\).
If \(a > 0\), the parabola opens upwards like a smile; if \(a < 0\), it opens downwards like a frown. In our example, the coefficient \(a = 3\) is positive, so the parabola opens upwards.
Understanding the direction and position of the parabola helps in sketching it and knowing where it meets the x-axis鈥攊.e., at the roots.
If \(a > 0\), the parabola opens upwards like a smile; if \(a < 0\), it opens downwards like a frown. In our example, the coefficient \(a = 3\) is positive, so the parabola opens upwards.
Understanding the direction and position of the parabola helps in sketching it and knowing where it meets the x-axis鈥攊.e., at the roots.
Vertex
The vertex of a parabola is its highest or lowest point, depending on whether it opens upwards or downwards. The coordinates of the vertex \((h, k)\) are calculated using:
- \(h = -\frac{b}{2a}\)
- \(k = ax^2 + bx + c\) evaluated at \(x = h\)
- \(h = -\frac{-3}{2 \times 3} = \frac{1}{2}\)
- Substituting \(h\) in the equation gives \(k = 3\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) - 1 = -\frac{7}{4}\)
Real Roots
Real roots of a quadratic function are the x-values where the parabola intersects the x-axis. The presence of real roots entirely depends on the discriminant:
- If \(\Delta > 0\), there are two distinct real roots.
- If \(\Delta = 0\), there is one real root, which means the parabola touches the x-axis at the vertex.
- If \(\Delta < 0\), there are no real roots, and the parabola does not intersect the x-axis.
- \(x = \frac{3 + \sqrt{21}}{6}\)
- \(x = \frac{3 - \sqrt{21}}{6}\)
