Using the Desmos website calculator, sketch a graph of a piecewise function

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Using the Desmos website calculator, sketch a graph of a piecewise function. Write the domain in interval notation. y = x^2, -1 ≤ x ≤ 1 and y = 3x – 2, 1 ≤ x ≤ 3 Then choose your own functions and show them on the Desmos website calculator graph.

The Correct Answer and Explanation is:

To graph the given piecewise function on Desmos, follow these steps:

Step-by-Step Instructions:

  1. Open Desmos: Visit the Desmos website and open the calculator.
  2. Input the Piecewise Function:
    • For the first function y=x2y = x^2y=x2 with domain −1≤x≤1-1 \leq x \leq 1−1≤x≤1: iniCopyEdity = {x^2, -1 ≤ x ≤ 1}
    • For the second function y=3x−2y = 3x – 2y=3x−2 with domain 1≤x≤31 \leq x \leq 31≤x≤3: iniCopyEdity = {3x - 2, 1 ≤ x ≤ 3}
    You should now see the piecewise function graphed on the Desmos plot.

Domain in Interval Notation:

  • The domain of the entire piecewise function is the union of the domains of the two individual pieces.
    • For the first piece, the domain is [−1,1][-1, 1][−1,1].
    • For the second piece, the domain is [1,3][1, 3][1,3].

Thus, the domain of the piecewise function in interval notation is:
[−1,3][-1, 3][−1,3]

Explanation:

  1. Graphing y=x2y = x^2y=x2 on [−1,1][-1, 1][−1,1]:
    This part of the piecewise function represents a parabola. The graph of y=x2y = x^2y=x2 is a U-shaped curve that is symmetric about the y-axis. On the domain [−1,1][-1, 1][−1,1], the function will plot only the portion between x=−1x = -1x=−1 and x=1x = 1x=1.
  2. Graphing y=3x−2y = 3x – 2y=3x−2 on [1,3][1, 3][1,3]:
    The second part of the piecewise function is a linear function. The slope of the line is 3, and the y-intercept is -2. This means the line will slope upwards as you move from left to right, starting at x=1x = 1x=1 and ending at x=3x = 3x=3.
  3. Connecting the Two Pieces:
    At x=1x = 1x=1, the two pieces meet because y=x2y = x^2y=x2 at x=1x = 1x=1 equals 1, and y=3x−2y = 3x – 2y=3x−2 at x=1x = 1x=1 also equals 1. Therefore, there is no jump or discontinuity between the two pieces at x=1x = 1x=1.

Choosing My Own Functions:

You could choose other functions to display as well. For example:

  1. Let y=x2y = x^2y=x2 on the domain [−2,0][-2, 0][−2,0].
  2. Let y=4x+1y = 4x + 1y=4x+1 on the domain [0,2][0, 2][0,2].

On Desmos, you would input:

 = {x^2, -2 ≤ x ≤ 0}
y = {4x + 1, 0 ≤ x ≤ 2}

This would plot a parabola and a line, with a continuous connection at x=0x = 0x=0

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