Parabola Transformation: Finding G(-4) After Reflection & Rotation

Let's break down this transformation problem step by step. We're starting with a parabola, reflecting it, rotating it, and then trying to find a specific value of the transformed function. Buckle up, because here we go!

Understanding the Initial Parabola

First, let's get to know our initial parabola, f(x) = x^2 - 4x - 12. Understanding its characteristics will help us visualize what happens during the transformations. We can rewrite this equation in vertex form to easily identify the vertex of the parabola. Completing the square, we have:

f(x) = (x - 2)^2 - 16

From this form, we can see that the vertex of the parabola is at the point (2, -16). The parabola opens upwards since the coefficient of the x^2 term is positive. Knowing the vertex and the direction the parabola opens is super useful for visualizing the reflection and rotation.

To further understand the initial function, we can identify the x-intercepts by setting f(x) = 0:

x^2 - 4x - 12 = 0 (x - 6)(x + 2) = 0

Thus, x = 6 or x = -2. This means the parabola crosses the x-axis at the points (6, 0) and (-2, 0). These points will also be transformed during the reflection and rotation. Having these key points—the vertex and the x-intercepts—gives us a good handle on the shape and position of the original parabola.

Think of it like this: we're taking a snapshot of this parabola, then moving that snapshot around according to the reflection and rotation rules. The better we understand the initial snapshot, the easier it will be to see where it ends up.

Reflection About y = 2

Next, we will reflec the parabola across the horizontal line y = 2. When reflecting a function across a horizontal line y = k, the transformation affects the y-coordinate of each point on the graph. The new y-coordinate, y', is given by:

y' = 2k - y

In our case, k = 2, so the transformation becomes:

y' = 4 - y

This means we're replacing every y with 4 - y in the equation of the parabola. So, if our original function is y = x^2 - 4x - 12, the reflected function, let's call it f'(x), becomes:

4 - y' = x^2 - 4x - 12

Solving for y', we get:

y' = -x^2 + 4x + 16

So, f'(x) = -x^2 + 4x + 16. Notice how the sign of the x^2 term has changed, indicating that the parabola now opens downwards. Also, the vertex has shifted. To find the new vertex, we can complete the square again:

f'(x) = -(x - 2)^2 + 20

The new vertex is at (2, 20). Reflecting the original vertex (2, -16) across the line y = 2 confirms this: 4 - (-16) = 20.

Remember those x-intercepts we found earlier? Let's see where they go after the reflection. The original x-intercepts were at (6, 0) and (-2, 0). After reflection, their y-coordinates become 4 - 0 = 4, so they are now at (6,4) and (-2,4). However, these are no longer x-intercepts because the y-coordinates are not zero. To find the new x-intercepts we would need to set f'(x) = 0 and solve for x. However, for this problem, we don't need the x-intercepts for this step.

Rotation of 90° About the Origin

Next, we are rotating the reflected parabola 90° counterclockwise around the origin (0, 0). When we rotate a point (x, y) 90° counterclockwise about the origin, the new coordinates (x', y') are given by:

x' = -y y' = x

This means we need to replace x with y' and y with -x' in the equation of the reflected parabola, f'(x) = -x^2 + 4x + 16. Replacing x with y' and y with -x', we get:

-x' = -(y')^2 + 4(y') + 16

Multiplying by -1, we have:

x' = (y')^2 - 4(y') - 16

Since g(x) represents the final transformed function, we can write:

g(x) = x^2 - 4x - 16

Notice how the rotation effectively swaps the roles of x and y (with a sign change), changing the orientation of the parabola. Now the parabola opens to the right, instead of up or down.

To see how the vertex changes, we take the vertex of the reflected parabola f'(x) which is (2, 20) and rotate it 90° counterclockwise around the origin, we get (-20, 2). We can also rewrite the equation for g(x) in vertex form:

g(x) = (x - 2)^2 - 20

Which means the vertex is at (2, -20). Because the roles of x and y are swapped, we have to be extra careful when interpreting the vertex and intercepts.

Finding g(-4)

Finally, we need to find the value of g(-4). We now have the equation for g(x):

g(x) = x^2 - 4x - 16

Substitute x = -4 into the equation:

g(-4) = (-4)^2 - 4(-4) - 16 g(-4) = 16 + 16 - 16 g(-4) = 16

Therefore, the value of g(-4) is 16.

So, the answer to this transformation journey is g(-4) = 16. We took the original parabola, reflected it across a horizontal line, rotated it, and then evaluated the resulting function at a specific point. Each step involved understanding how the transformations affect the equation and the key features of the parabola. Great job, guys!