Understanding the X- and Y-Intercepts of a Quadratic Equation

The equation y 3x2 is a simple quadratic equation. In this article, we will explore how to find its x-intercept and y-intercept.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, meaning the highest exponent of the variable (in this case, x) is 2. The general form of a quadratic equation is y ax2 bx c. In our specific example, the equation is y 3x2, where the coefficient a is 3, b is 0, and c is 0.

Y-Intercept

The y-intercept is the point at which the graph of a function intersects the y-axis. This occurs when x 0.

Substituting x 0 into the equation:

y 3(0)2

y 0

Therefore, the y-intercept is at the point (0, 2). This is because the original equation was y 3x2 2, which simplifies to y 3(0)2 2, resulting in y 2.

X-Intercept

The x-intercept is the point at which the graph of a function intersects the x-axis. This occurs when y 0.

To find the x-intercept, we set y 0 and solve for x in the equation:

0 3x2 2

First, we isolate the term with x by subtracting 2 from both sides:

-2 3x2

Then, we divide both sides by 3:

x2 -2/3

Solving for x gives:

x ±√(-2/3)

However, since the square root of a negative number is not real, it indicates that there are no real x-intercepts for this equation.

Summary

Y-Intercept: (0, 2)
X-Intercept: None (No real x-intercepts)

Additional Information

The x-intercept of a quadratic equation in the form y 3x2 doesn't exist as a real number because the value under the square root is negative. This is in contrast to linear equations, where you can have one or two x-intercepts, depending on the value of the constant term.

Understanding the intercepts of a quadratic equation is crucial for graphing the function. In the case of y 3x2 2, the y-intercept is at (0, 2) and there are no real x-intercepts.