How to Rotate a Graph Equation?

To rotate a graph equation by an angle θ, substitute x with x cos θ - y sin θ and y with x sin θ + y cos θ in the original equation.

Here's a breakdown of how to perform this rotation:

1. Identify the Angle of Rotation: Determine the angle, denoted as θ (theta), by which you want to rotate the graph.

2. Perform the Substitutions: Everywhere you see x in the original equation, replace it with (x cos θ - y sin θ). Everywhere you see y, replace it with (x sin θ + y cos θ).

3. Simplify the Equation: After the substitutions, simplify the resulting equation algebraically. This may involve expanding terms, combining like terms, and using trigonometric identities. The simplified equation represents the rotated graph.

Example:

Let's say you have the equation y = x (a straight line) and you want to rotate it by 45 degrees (θ = 45° or π/4 radians).

• cos(45°) = √2 / 2
• sin(45°) = √2 / 2

Now, perform the substitutions:

• Replace x with x cos(45°) - y sin(45°) = (x√2 / 2) - (y√2 / 2)
• Replace y with x sin(45°) + y cos(45°) = (x√2 / 2) + (y√2 / 2)

The original equation y = x becomes:

(x√2 / 2) + (y√2 / 2) = (x√2 / 2) - (y√2 / 2)

Simplifying, we get:

y√2 / 2 = -y√2 / 2
Which implies
2y√2 / 2 = 0
So y = 0.

Therefore, the original line rotated 45 degrees becomes the x-axis.

Summary:

Rotating a graph equation involves a systematic substitution process using trigonometric functions (sine and cosine) of the rotation angle. This method allows you to manipulate and analyze the graph's orientation on the coordinate plane. Remember to carefully simplify the equation after substitution to obtain the final representation of the rotated graph.