Understanding and Finding the Y-Intercept of the Quadratic Equation y -3x2 12x - 7

Quadratic equations are a fundamental part of algebra, and understanding their properties is crucial for a wide range of applications, from physics to engineering. In this article, we will delve into the concept of the y-intercept of a quadratic equation and provide a detailed step-by-step guide on how to find it for the specific equation y -3x2 12x - 7. We will also explore the role of graphing calculators and other tools in aiding the process.

What is the Y-Intercept?

The y-intercept of a quadratic equation is the point where the graph of the equation intersects the y-axis. This point is unique because it has a specific characteristic: the x-coordinate is always zero. In other words, when x 0, we can determine the value of y, which gives us the coordinates of the y-intercept.

Finding the Y-Intercept of y -3x2 12x - 7

To find the y-intercept of the quadratic equation y -3x2 12x - 7, we need to substitute x 0 into the equation. This is because the y-axis is defined by x 0. Let's go through the steps:

Step 1: Substitute x 0 in the equation y -3x2 12x - 7.

Step 2: Simplify the equation.

Step 3: Determine the y-coordinate of the y-intercept.

Substitute x 0 in the equation: y -3(0)2 12(0) - 7 Simplify the equation step by step: 0 0 - 7 -7

Therefore, the y-intercept of the equation y -3x2 12x - 7 is (0, -7).

Graphical Interpretation

While we have found the y-intercept algebraically, plotting the equation on a graph using tools like Desmos Graphing Calculator can provide a visual confirmation. When using Desmos, the equation y -3x2 12x - 7 will intersect the y-axis at the point (0, -7).

Using the Quadratic Formula

Beyond the y-intercept, the quadratic formula can also be used to find the x-intercepts, which are the points where the graph touches or crosses the x-axis. The quadratic formula is given by:

Quadratic Formula:

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

For the equation y -3x2 12x - 7, the coefficients are:

a -3 b 12 c -7

Let's calculate the x-intercepts:

calculate (b^2 - 4ac): 122 - 4(-3)(-7) 144 - 84 60 Now, use the quadratic formula: [x frac{-12 pm sqrt{60}}{-6}] The calculation yields two solutions: [x frac{-12 sqrt{60}}{-6} approx 0.707] [x frac{-12 - sqrt{60}}{-6} approx 3.291] The x-intercepts are approximately 0.707 and 3.291.

Conclusion

In conclusion, the y-intercept of the quadratic equation y -3x2 12x - 7 is (0, -7). This point is significant and provides valuable information about the behavior of the quadratic function. Understanding these concepts is essential for advanced algebraic and analytical tasks, and tools like graphing calculators are invaluable in enhancing comprehension and accuracy.

For further exploration, you can utilize graphing calculators like Desmos and explore the complete graph of the equation.