Question

INTERPRETING THE DISCRIMINANT Consider the equation \(\frac{1}{2} x^{2}+\frac{2}{3} x-3=0\) What does the discriminant tell you about the graph of \(y=\frac{1}{2} x^{2}+\frac{2}{3} x-3 ?\) Does the graph cross the \(x\) -axis?

Short answer

By calculating and analyzing the discriminant of the equation, a determination can be made about whether, and at how many points, the graph of the equation \(y = \frac{1}{2}x^{2} + \frac{2}{3}x - 3\) crosses the x-axis.

Step-by-step solution

Identify a, b, and c from the equation

The given quadratic equation can be written in the standard form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. Here, \(a = \frac{1}{2}\), \(b = \frac{2}{3}\), and \(c = -3\).

Calculate the discriminant

The discriminant of a quadratic equation is given by the expression \(D = b^{2} - 4ac\). Substituting \(a = \frac{1}{2}\), \(b = \frac{2}{3}\), and \(c = -3\) into this expression, the discriminant is calculated as \(D = (\frac{2}{3})^{2} - 4 \cdot \frac{1}{2} \cdot -3\). The resulting value of \(D\) will determine the nature of the roots of the equation.

Analyze the solutions based on the discriminant

The solutions to the original equation, and thus the points where the graph of \(y = \frac{1}{2}x^{2} + \frac{2}{3}x - 3\) intersects the x-axis, correspond to the roots of the equation. We can interpret how many roots the equation has and hence how many points of intersection there are with the x-axis based on the value of the discriminant. If \(D > 0\), the equation has two real and distinct roots, hence the graph intersects the x-axis at two distinct points. If \(D = 0\), the equation has one real root (a repeated root), hence the graph touches the x-axis at one point. If \(D < 0\), there are two complex roots, hence the graph does not intersect the x-axis.

Key concepts

Quadratic Equations

Quadratic equations are a staple in algebra and represent a fundamental type of polynomial equation. A quadratic equation is any equation that can be written in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to zero. This standard form reveals key features of the equation's graph, such as its curvature, orientation, and whether it opens upwards or downwards.

For example, in the equation \( \frac{1}{2} x^2 + \frac{2}{3} x - 3 = 0 \), \( a \) is positive, which means the graph of this quadratic function opens upwards. These equations are vital in various areas, including physics, engineering, finance, and many more, reflecting their importance in understanding complex patterns.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a powerful tool that gives you insight into the nature of its roots without actually solving the equation. It is calculated with the formula \( D = b^2 - 4ac \), using the coefficients from the standard form \( ax^2 + bx + c = 0 \).

Interpreting the value of the discriminant is straightforward:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( D < 0 \), the roots are complex and come in a pair of complex conjugates.

This information provides a quick way to discern how a quadratic function will behave with respect to the x-axis and advise on the potential solutions or points of intersection with the x-axis.

Graph of a Quadratic Function

The graph of a quadratic function is a parabola. In a coordinate plane, this curve has a distinctive 'U' shape, which can open either upwards or downwards based on whether the coefficient \( a \) is positive or negative, respectively. The peak or valley point of a parabola is called the vertex, and this point represents the maximum or minimum value of the quadratic function.

Moreover, the axis of symmetry, which is a vertical line that passes through the vertex, divides the parabola into mirror images. The intersection points of the parabola with the x-axis are known as the roots or zeroes of the quadratic function. These roots correspond to the solutions of the quadratic equation derived from setting the function equal to zero.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions to the equation when set equal to zero and represent the x-values where the graph of the quadratic function intersects the x-axis. There can be one, two, or no real roots depending on the discriminant's value. With our example, \( \frac{1}{2} x^2 + \frac{2}{3} x - 3 = 0 \), calculating the discriminant helps us predict the presence and number of these intersections without requiring us to solve the equation.

If a quadratic equation has real roots, these roots can also be thought of as the x-coordinates of the points where the graph of the corresponding quadratic function touches or crosses the x-axis. These points are significant as they can give us an understanding of the structure of the function and the behavior of the related physical or theoretical system it models.