Sketch a graph of f(x)=-2|x-1|+2. Before sketching the graph, determine where the function has its minimum or maximum value so you can place your first point there.

Answer 1

To find the minimum or maximum value of the function f(x)=-2|x-1|+2, we need to analyze the behavior of the absolute value function |x-1|.

The expression inside the absolute value, x-1, is the linear function of x. When x is to the left of 1, the expression evaluates to a negative value, giving us -(negative) = positive. When x is to the right of 1, the expression evaluates to a positive value, resulting in a negative output: -(positive) = negative.

Thus, the graph of the absolute value function will have a "V" shape with the vertex at the point (1, 0), which represents the minimum value.

Now let's plot the graph:

1) Place the vertex (1, 0) as the first point on the graph.

2) Since -2 is multiplied by the absolute value function, the graph will be inverted and vertically stretched by a factor of 2. This means that every point on the graph will have a y-coordinate twice as large as the corresponding point on the absolute value function's graph.

3) Draw the "V" shape, starting from the vertex (1, 0), going downwards symmetrically. Remember that the y-values will be multiplied by -2 and it will be translated upwards by 2 units due to the "+2" in the function.

This is how the graph will look like:

|
| /
2 | /
| /
| /
0 | /
|/
-2 ---|-----|----|----|----|----|----
-2 0 1 2 3 4 5

Thus, the minimum value of the function f(x) is 0, and it occurs at x = 1.