Problem 43 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The graph of the function intersects the x-axis at exactly one point.

Step 1: Identify the coefficients

In the equation \(y = 3x^{2} - 6x + 3\), the coefficients are \(a = 3\), \(b = -6\), and \(c = 3\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \(ax^{2} + bx + c = 0\) is given by \(D = b^{2} - 4ac\). Substituting the coefficients into the formula gives \(D = (-6)^{2} - 4 * 3 * 3 = 36 - 36 = 0\).

Step 3: Determine the number of roots

Since the discriminant is zero, the quadratic equation has exactly one root. Therefore, the graph of the function intersects the x-axis at exactly one point.

Key Concepts

Discriminant

The discriminant is a helpful tool in understanding the nature of the roots of a quadratic equation. It's derived from the general form of a quadratic equation: \(ax^2 + bx + c = 0\). By using the formula \(D = b^2 - 4ac\), you can determine how many times the graph of the quadratic equation will intersect the x-axis.

• If \(D > 0\), there are two distinct real roots, meaning the graph intersects the x-axis at two separate points.
• If \(D = 0\), there is exactly one real root. This situation is known as a repeated root or a double root, and it means the graph touches the x-axis at one point.
• If \(D < 0\), there are no real roots; instead, the roots are complex numbers, and the graph does not intersect the x-axis at all.

This concept is particularly useful in predicting the behavior of the graph without having to graph it manually. In the original exercise, because the discriminant is exactly zero, we know the quadratic's single intersection with the x-axis.

Graphing Quadratics

Graphing quadratic equations can visually demonstrate the function's properties, such as its vertex, axis of symmetry, and the number of x-axis intersections. A quadratic graph forms a parabola, which can open upwards or downwards depending on the coefficient of \(x^2\).To graph a quadratic equation efficiently:

• Identify the vertex, which is the highest or lowest point on the graph. The vertex can be calculated using \((-b/2a, f(-b/2a))\).
• Determine the axis of symmetry, which is a vertical line through the vertex, given by \(x = -b/2a\).
• Find the y-intercept, occurring where the graph intersects the y-axis (simply the constant term \(c\)).

By plotting these key features, you can sketch the parabola's shape. Understanding how these features and the discriminant impact the graph helps you predict the number of x-intercepts, like in the original exercise where the discriminant was zero, illustrating just one x-intercept.

Roots of Quadratic Equations

The roots (or solutions) of a quadratic equation are the x-values at which the quadratic graph intersects the x-axis. These roots represent the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).To find these roots, you can utilize several methods:

• Factoring, which involves expressing the quadratic as a product of binomials.
• The Quadratic Formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), particularly effective when the quadratic cannot be easily factored.
• Completing the Square, which rewrites the equation in vertex form to identify the roots.

Determining the discriminant first helps decide which method might be most efficient. In cases where the discriminant equals zero, like in our provided problem, the quadratic formula will show that both potential solutions are the same, indicating a single root where the graph just touches the x-axis at one point.