Problem 12 - Algebra 1: Concepts and Skills

Question

Determine whether the graph of the function will intersect the x-axis in zero, one, or two points. \(y=6 x-3-3 x^{2}\)

Step-by-Step Solution

Answer

The graph of the function \(y = 6x -3 -3x^2\) intersects the x-axis at exactly one point.

Step 1: Rewrite the equation in the standard quadratic form

Rearranging the equation \(y = 6x -3 -3x^2\), we obtain the standard quadratic form \(y = -3x^2 + 6x - 3\). Now, let's set the function equal to zero to obtain the equation \(-3x^2 + 6x - 3 = 0\). Now, we can identify the coefficients \(a = -3\), \(b = 6\), and \(c = -3\).

Step 2: Compute the discriminant

The discriminant for a quadratic equation is given by \(D = b^2 - 4ac\). Substituting our coefficients for a, b, and c into this equation, we get \(D = (6)^2 - 4*(-3)*(-3) = 36 - 36 = 0\).

Step 3: Determine the number of solutions

Since the discriminant D = 0, the quadratic equation has exactly one real solution. Thus, the graph of the function intersects the x-axis exactly at one point.

Key Concepts

Discriminant of a Quadratic Equation

Understanding the discriminant of a quadratic equation is vital for predicting the nature of its solutions. The discriminant, denoted usually as D, is part of the quadratic formula and is calculated from the coefficients of the quadratic equation in its standard form, which is usually given as \(ax^2 + bx + c = 0\). It is expressed mathematically as \(D = b^2 - 4ac\).

The value of the discriminant provides us with critical information:

• If \(D > 0\), the quadratic equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, which means the graph of the function will touch the x-axis at one point (this point is also known as a double root).
• If \(D < 0\), the equation has no real solutions; instead, there are two complex solutions.

In our exercise, setting the function equal to zero allows us to use the discriminant to determine the number of times the graph intersects the x-axis. With \(D=0\) in our example, we can conclude that there is just one intersection point.

Standard Quadratic Form

The standard form of a quadratic equation is the starting point for analyzing its behavior and is expressed as \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The graph of a quadratic function is a parabola, and the sign of the coefficient \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

In our example, rearranging the equation to fit this form reveals that \(a = -3\), which indicates the parabola opens downwards. The conversion to the standard form not only allows us to utilize the discriminant but also serves to identify the vertex, axis of symmetry, and direction of the parabola's opening.

Quadratic Equations Solutions

Solving a quadratic equation entails finding the values of \(x\) that make the equation true (i.e., the solutions). These are the points where the graph of the quadratic function intersects the x-axis. Solutions can be found using a variety of methods such as factoring, completing the square, graphically, or using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\).

The discriminant, as discussed earlier, informs us about the nature and number of solutions. For example, a discriminant of zero implies that there is one real solution to the equation, which means our quadratic function intersects the x-axis at a single point. This solution can also be referred to as the 'root' of the equation. In more complex cases where \(D > 0\) or \(D < 0\), we would use the quadratic formula to calculate the exact solutions.