Problem 10 - Algebra 1: Concepts and Skills

Question

Determine whether the graph of the function will intersect the x-axis in zero, one, or two points. \(y=x^{2}+2 x+4\)

Step-by-Step Solution

Answer

The function does not have any x-intercepts (it does not cross the x-axis at any point).

Step 1: Write the Function in Standard Form

The function \(y=x^{2}+2 x+4\) is already in the standard quadratic form \((ax^{2} + bx + c)\) where a=1, b=2 and c=4. Hence, there is no need of rewriting the equation.

Step 2: Successively Calculate the Discriminant

Calculate the value of the discriminant using the formula \(b^{2}-4ac\). Substituting the values of a=1, b=2 and c=4, we find that the discriminant is \(2^{2} - 4*1*4 = 4 - 16 = -12\).

Step 3: Interpret the Result

Since the discriminant is negative (-12), it means the function does not intersect the x-axis. Therefore, the answer to the original problem is zero.

Key Concepts

Discriminant

The discriminant is a critical part of a quadratic equation that determines the nature of its roots. It is derived from the quadratic formula, which helps in finding the roots of any quadratic equation.The formula for the discriminant is given by:\[ b^2 - 4ac \]where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation in its standard form, \( ax^2 + bx + c \). The value of the discriminant tells us:

• If it's positive, the quadratic function has two distinct real roots.
• If it's zero, there is exactly one real root, also known as a repeated or double root.
• If it's negative, the function has no real roots, meaning it does not cross the x-axis.

In this exercise, the discriminant calculated is \(-12\), indicating that the graph of the function will not intersect the x-axis at any point.

Standard Form

The standard form of a quadratic equation is the format \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This is the most common way to express a quadratic equation and is essential for various computations, including finding the discriminant. In this exercise, the function \( y = x^2 + 2x + 4 \) is already presented in standard form:

• \( a = 1 \)
• \( b = 2 \)
• \( c = 4 \)

By having the equation in this form, it becomes straightforward to substitute these values into formulas, like the discriminant, to examine the properties of the function.

Graph Intersection

Graph intersection with the x-axis involves determining where a function crosses or touches this axis. For quadratic functions, this is analyzed by solving the equation \( y = 0 \) or, equivalently, \( ax^2 + bx + c = 0 \). The roots (or solutions) of this equation give the x-coordinates where the graph intersects the x-axis. However, the discriminant plays a key role in determining how many such intersections exist:

• A positive discriminant means two intersections.
• A zero discriminant results in a single intersection, where the vertex of the parabola just touches the axis.
• A negative discriminant indicates no intersections.

In our case, the discriminant is \(-12\); thus, the parabola formed by the function \( y = x^2 + 2x + 4 \) does not intersect with the x-axis. This can be visually interpreted as the parabola being above the x-axis without touching or crossing it.