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.
To determine 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 function, (x - 1), represents the distance between x and 1 on the number line. The absolute value function takes this distance and returns its positive value.
Since the absolute value function will always return a non-negative value, the expression -2|x-1| will be non-positive for all values of x.
Therefore, the function f(x) = -2|x-1| + 2 will have its maximum value when -2|x-1| is equal to 0. This occurs when x-1 is equal to 0, or x = 1.
So the first point on the graph of f(x) will be when x = 1.
Now, let's plot this point and sketch the graph.
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:
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2 | /
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0 | /
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-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.
To determine the minimum or maximum value of the function f(x) = -2|x-1| + 2, we need to find the critical points where the slope changes. Since the absolute value function |x-1| changes its sign at x = 1, we have two cases to consider:
1. For x < 1: In this case, the expression |x-1| is equivalent to -(x-1), which simplifies to -x + 1. Therefore, f(x) = -2(-x + 1) + 2 = 2x.
2. For x ≥ 1: In this case, |x-1| is equivalent to (x-1). Therefore, f(x) = -2(x-1) + 2 = -2x + 4.
Now, let's plot the two cases separately on a graph:
For x < 1:
- The slope of f(x) = 2x is positive, indicating that the graph increases as x moves to the right.
- To determine the first point, we can substitute x = 1 into the expression. f(1) = 2*1 = 2.
- Therefore, the first point is (1, 2).
For x ≥ 1:
- The slope of f(x) = -2x + 4 is negative, indicating that the graph decreases as x moves to the right.
- To determine the first point, we can substitute x = 1 into the expression. f(1) = -2*1 + 4 = 2.
- Therefore, the first point is (1, 2).
Using the critical point (1, 2) as the starting point, you can now sketch the graph of f(x) = -2|x-1| + 2 by incorporating the information about the slopes and the critical point.