Vertex Form of Quadratic Equations: Understanding 3 Left and 4 Down

In quadratic equations, vertex form is particularly useful for identifying key features such as the vertex (the highest or lowest point of the parabola) and the direction of the parabola's opening. The transformation "3 left and 4 down" refers to horizontal and vertical shifts of the graph. Understanding these transformations helps in visualizing and analyzing the graph of a quadratic equation more effectively.

Vertex of the Parabola and Its Transformation

The vertex of a parabola in vertex form (y a(x - h)^2 k) is the point (h, k). In the equation given, y 4(x - 8)^2 - 3, the vertex of the parabola is (8, -3). The transformation "3 left and 4 down" manipulates the vertex of the parabola to a new location.

Manipulating the Vertex

To understand the transformation, let's break it down:

3 left: This is a horizontal shift. A left shift is indicated by a negative value in the equation. If we move 3 units to the left, the x-coordinate of the vertex, 8, will become 8 - 3, which equals 5. 4 down: This is a vertical shift. A downward shift is indicated by subtracting the value from the y-coordinate. If we move 4 units down, the y-coordinate, -3, will become -3 - 4, which equals -7.

Therefore, the new position of the vertex after the transformation "3 left and 4 down" is (5, -7).

Transformed Equation

The given equation is y 4(x - 8)^2 - 3. After applying the transformation, the equation becomes:

y 4(x - 5)^2 - 7

This new equation reflects the transformed vertex (5, -7).

Steps to Find the New Equation

To find the new equation after the transformation "3 left and 4 down," follow these steps:

Identify the vertex of the original equation: (8, -3). Apply the horizontal shift for "3 left": 8 - 3 5. Apply the vertical shift for "4 down": -3 - 4 -7. Substitute the new vertex (5, -7) into the vertex form of the equation.

The new equation after transformation is:

y 4(x - 5)^2 - 7

Conclusion

Understanding the concept of "3 left and 4 down" is crucial for transforming quadratic equations and visualizing their graphs. This concept helps in predicting the movement of the vertex and the overall shape of the parabola. By mastering these transformations, you can effectively analyze and solve quadratic equations and related problems in algebra.