Problem 51 - Algebra 1: Concepts and Skills

Question

NUMBER OF X-INTERCEPTS Determine whether the graph of the function intersects the \(x\) -axis in zero, one, or two points. $$ y=x^{2}+2 x+15 $$

Step-by-Step Solution

Answer

The graph of the function does not intersect the X-axis at any point.

Step 1: Write the function in standard form

The function is already in standard form, which is in the format \(ax^2 + bx + c = 0\). Where a, b, and c are coefficients. Here \(a = 1\), \(b = 2\), \(c = 15\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is calculated by the formula \(D = b^2 - 4ac\). Using the formula, we find \(D = 2^2 - 4*1*15 = 4 - 60 = -56.\)

Step 3: Analyze the discriminant

The discriminant value \(D = -56\) is negative. In the quadratic equation, if the discriminant is negative, it means there are no real roots. Since the x-intercepts are the real roots of the equation, it indicates that the graph of the function does not intersect the X-axis at any point.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function characterized by the highest exponent being a square, or 2. The general form of a quadratic function is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

These functions graph as parabolas and can open upwards or downwards depending on the sign of \(a\). If \(a > 0\), the parabola opens upwards, whereas if \(a < 0\), it opens downwards.

Quadratic functions are found in various real-world scenarios, such as projectile motion, where they model the path of an object under gravity without considering other forces such as air resistance.

To explore their properties, a helpful starting point is the standard form, which neatly organizes everything needed to use the formula for finding x-intercepts.

Discriminant

The discriminant is a critical part of solving quadratic equations because it gives valuable insight into the nature of the roots. It is derived from the quadratic formula and is calculated as \( D = b^2 - 4ac \).

The discriminant tells us the number of real roots, and hence, how many x-intercepts exist for a quadratic function:

• If \(D > 0\), the quadratic has two distinct real roots. This means the parabola will cross the x-axis at two points.
• If \(D = 0\), there is exactly one real root, or a repeated root, and the parabola touches the x-axis at just one point.
• If \(D < 0\), there are no real roots. The parabola doesn't intersect the x-axis, meaning it lies entirely above or below it depending on whether it opens upwards or downwards.

This concept is helpful not just in algebraic exercises but also in physics and engineering because it analyzes scenarios for feasibility and stability.

Real Roots

Real roots of a quadratic equation are the points where the graph intersects the x-axis, known as x-intercepts. These are the solutions of the equation when it is set to zero.

Understanding the nature of these roots is essential because they reflect the solutions of many real-world problems. For a quadratic to have real roots, the discriminant must be non-negative:

• If there are two real roots, it implies that there's a symmetrical crossing of the graph on the x-axis.
• If there's one real root, it's a tangent point where the parabola just touches the axis.
• Without real roots, the function has no intersection with the x-axis, indicating it might model a situation where no solution or interception is possible, such as no braking point in a given motion.

Hence, determining the real roots is an indispensable part of analyzing quadratic functions, as they offer practical insights into the function's behavior and potential intersections.