Problem 61 - Algebra 2

Question

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=3 x^{2}-10 x+6 $$

Step-by-Step Solution

Answer

The function has two distinct real x-intercepts.

Step 1: Identify the type of function

The given equation represents a quadratic function in the form of \(y=ax^2+bx+c\), where \(a=3\), \(b=-10\), and \(c=6\). Quadratic functions are parabolic and can have either 0, 1, or 2 x-intercepts.

Step 2: Calculate the Discriminant

The discriminant of a quadratic function, given by \(D=b^2-4ac\), determines the number of x-intercepts. For the given function, substitute \(a=3\), \(b=-10\), and \(c=6\) into the discriminant formula to get \(D=(-10)^2-4\cdot3\cdot6\).

Step 3: Interpret the Discriminant

Calculate the discriminant: \(D=100-72=28\). Since \(D>0\), the quadratic function has two distinct real x-intercepts.

Key Concepts

Quadratic Functions

Quadratic functions, like the one in our exercise, form the foundation of algebra and appear frequently in various mathematical contexts. Their standard form is given by the equation
\(y = ax^2 + bx + c\),
where \(a\), \(b\), and \(c\) are constants, and \(a ≠ 0\). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient \(a\).

These functions are ubiquitous in physics, engineering, and economics, modeling phenomena such as projectile motion and profit maximization.

Understanding the properties of quadratic functions, such as the axis of symmetry, vertex, and direction of opening, allows students to predict the behavior of the parabola. Moreover, these functions can have zero, one, or two real x-intercepts—the points where the graph crosses the x-axis, also known as the solutions of the equation. The nature of these intercepts can be revealed through an analysis of the discriminant.

Discriminant Analysis

Determining the number of x-intercepts a quadratic function has without graphing involves discriminant analysis—a critical concept in quadratic equations. The discriminant is the part of the quadratic formula beneath the square root sign and is given by \(D=b^2 - 4ac\).

This value tells us a lot about the nature of the roots:

• If \(D > 0\), the equation has two distinct real roots; the parabola crosses the x-axis at two points.
• If \(D = 0\), the equation has exactly one real root; the parabola touches the x-axis at one point, known as the vertex.
• If \(D < 0\), the equation has no real roots; the parabola doesn't cross the x-axis at all.

By substituting our values of \(a\), \(b\), and \(c\) into the discriminant formula, we can predict the number of x-intercepts or real solutions. In the given exercise, the discriminant was positive, indicating two real x-intercepts for our function.

Parabola Graph Characteristics

The graph of a quadratic function is known as a parabola. Parabolas have distinct characteristics that allow for visual interpretation and understanding of the function's behavior. For instance, the vertex is the highest or lowest point on the graph, serving as a pivot point. The axis of symmetry is a vertical line that divides the parabola into two mirror images, passing through the vertex.

The direction in which a parabola opens—upwards or downwards—is determined by the sign of the coefficient \(a\). When \(a > 0\), the parabola opens upwards, and the vertex represents the minimum point. Conversely, if \(a < 0\), it opens downwards, and the vertex is the maximum point.

Additionally, parabolas have intercepts: x-intercepts (where the graph crosses the x-axis) and a y-intercept (where it crosses the y-axis).

Interpreting x-intercepts

For the quadratic function in our exercise, the discriminant told us that there are two x-intercepts. Visually, this means that we're looking at a parabola that crosses the x-axis at two distinct points. Understanding these graph characteristics helps in grounding abstract concepts and facilitates a more intuitive learning experience.