f(x) = |x – 3| – 1
To better understand the function f(x) = |x – 3| – 1, let’s break it down step by step:
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To better understand the function f(x) = |x – 3| – 1, let’s break it down step by step:
1. Absolute Value: The absolute value of a number is always positive and represents the distance of that number from zero on a number line. As indicated by the vertical bars surrounding the expression (x – 3), this means we’ll be working with the distance between x and 3.
2. Difference: The expression (x – 3) represents the difference between x and 3. If x is greater than 3, the expression evaluates to a positive value. If x is less than 3, the expression evaluates to a negative value.
3. Absolute Value of (x – 3): The absolute value of (x – 3) ensures that the result is always positive, regardless of whether (x – 3) is positive or negative.
4. Subtract 1: Finally, subtracting 1 from the absolute value of (x – 3) shifts the entire graph down by 1 unit.
Now, let’s look at the graph of f(x) = |x – 3| – 1:
1. Graph the basic linear function y = x: This is a straight line with a slope of 1 and passing through the origin.
2. Apply the absolute value: For x > 3, the graph remains the same. However, for x < 3, the graph reflects or "flips" over the x-axis. 3. Shift the graph down by 1: Move the entire graph downward by 1 unit. Combining these steps, we obtain the graph of f(x) = |x - 3| - 1.
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