Understanding the X-Intercepts of Quadratic Functions: A Practical Example and Advanced Techniques

Quadratic functions are fundamental in mathematics, enabling us to model a variety of real-world phenomena. These functions take the form of a parabola and are defined by the equation: y ax^2 bx c, where a, b, and c are constants. One of the key points of interest in a quadratic function is its x-intercept, which is determined when y 0. This article will guide you through the process of finding the x-intercepts for a specific quadratic function, and introduce an alternative method using the quadratic formula.

An Initial Approach to Finding X-Intercepts

Let's consider a specific quadratic function: y -x^2 1. Our goal is to find the x-values for which y 0.

To begin, we start by setting the function equal to zero:

-x^2 1 0

Next, we rearrange this equation so that all terms are on one side, making the equation standard form:

x^2 - 1 0

To solve this, we can factor the left-hand side of the equation:

(x 1)(x - 1) 0

This factorization yields two cases:

Case 1: Solving x 1 0

x 1 0

x -1

Case 2: Solving x - 1 0

x - 1 0

x 1

Thus, the x-intercepts of the function y -x^2 1 are x -1 and x 1.

To verify these answers, we substitute the x-values back into the original function:

For x -1:

y -(-1)^2 1 -1 1 0

For x 1:

y -(1)^2 1 -1 1 0

These results confirm that the x-intercepts are correct.

Exploring the Quadratic Formula for X-Intercepts

Another method for finding the x-intercepts involves using the quadratic formula, which is derived from the general form of a quadratic function: ax^2 bx c 0. The quadratic formula is:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

For our function y -x^2 1, we identify the coefficients:

a -1 b 0 c 1

Substituting these values into the quadratic formula, we get:

x frac{-0 pm sqrt{0^2 - 4(-1)(1)}}{2(-1)}

Which simplifies to:

x frac{pm sqrt{4}}{-2}

This further simplifies to:

x frac{pm 2}{-2} -1 text{ or } 1

Hence, using the quadratic formula, we again find that the x-intercepts are -1 and 1.

Conclusion

Both the factoring method and the quadratic formula can be used to find the x-intercepts of a quadratic function. For the quadratic function y -x^2 1, the x-intercepts are x -1 and x 1. This example demonstrates the practical application of these techniques and their consistency in providing accurate results.

Further Reading Resources

MathIsFun: Quadratic Equation Khan Academy: Finding X-Intercepts Purplemath: Advanced Quadratic Formula