Transformation of ty then transformations of the graphs y=x^2or y=|x| to sketch a graph of f by hand.

f(x)=2|x-6|-4

To sketch the graph of the function f(x) = 2|x-6| - 4, we will first start with transforming the graph of y = x^2 (a basic quadratic function), and then transform it by applying the necessary translations and reflections.

Step 1: Start with the graph of y = x^2.
- Plot some points on the graph of y = x^2 to create a basic parabolic shape. For example, you can pick some x values (e.g., -3, -2, -1, 0, 1, 2, 3) and calculate the corresponding y values by squaring them (e.g., (-3)^2 = 9, (-2)^2 = 4, (-1)^2 = 1, 0^2 = 0, 1^2 = 1, 2^2 = 4, 3^2 = 9).
- Connect the plotted points smoothly to create a parabolic curve.

Step 2: Transform the graph of y = x^2 to y = |x|.
- Apply the absolute value transformation by reflecting the portion of the graph below the x-axis to the corresponding portion above the x-axis.
- The vertex of the parabola (0, 0) remains the same after the transformation, so you don't need to move the vertex.

Step 3: Transform the graph of y = |x| to f(x) = 2|x-6| - 4.
- Start by applying a horizontal translation of 6 units to the left. This shifts the entire graph horizontally to the left by 6 units.
- Then, apply a vertical translation of 4 units downward. This shifts the entire graph downward by 4 units.

After performing these transformations, plot some new points on the transformed graph and connect them smoothly to form the final shape of the graph f(x) = 2|x-6| - 4.

- To plot the points, you can choose a few x values (e.g., 0, 3, 6, 9) and calculate the corresponding y values by substituting them into the function f(x) = 2|x-6| - 4.
- Connect the plotted points smoothly to create the final graph.

Remember that the key transformations involved in this process are:
1. Absolute value transformation
2. Horizontal translation
3. Vertical translation.

By following these steps, you should be able to sketch the graph of f(x) = 2|x-6| - 4 by hand.