Graph of the Step Function f(x) = [x + 3] over -2 ≤ x ≤ 2
Here's how to graph the step function f(x) = [x + 3] over the interval -2 ≤ x ≤ 2:
1. Identify the Step Levels:
The step function f(x) = [x + 3] takes different integer values depending on the input x. Within the given interval:
For x < -2, f(x) = [x + 3] = -2 (since any number less than -2 added to 3 will still be less than 2).
For -2 ≤ x ≤ -1, f(x) = [x + 3] = 2 (rounding -1 + 3 up to the nearest integer gives 2).
For x > -1, f(x) = [x + 3] = 3 (since any number greater than -1 added to 3 will be greater than 2).
2. Draw Horizontal Lines:
Draw a horizontal line at y = -2 for all x values between -2 and -1.
Draw a horizontal line at y = 2 for all x values between -1 and 0.
Draw a horizontal line at y = 3 for all x values between 0 and 2 (including 2).
3. Connect the Lines with Vertical Jumps:
At x = -1, draw a vertical jump from the line at y = -2 to the line at y = 2. This represents the sudden change in the function's value from -2 to 2 as x crosses -1.
At x = 0, draw another vertical jump from the line at y = 2 to the line at y = 3. This represents the change in value from 2 to 3 as x crosses 0.
4. Label the Axes:
Label the x-axis as "x."
Label the y-axis as "f(x)."
5. (Optional) Show Discontinuities:
If needed, you can mark the points -1 and 0 with small open circles to signify the discontinuities where the function jumps between different levels.
This will give you the complete graph of the step function f(x) = [x + 3] over the interval -2 ≤ x ≤ 2. Remember that step functions have distinct jumps at integer values where the function's output changes abruptly.