Complete The Equation For The Piecewise Function Graphed Below

Here's how you can determine the equation of a piecewise function from its graph. A piecewise function, as the name suggests, is defined by multiple sub-functions, each applicable over a specific interval of the domain.

Understanding Piecewise Functions

Piecewise functions are a fundamental concept in mathematics, allowing us to model situations where different rules apply under different conditions. Imagine a tax system where the percentage you pay changes based on your income bracket – that's a real-world example of a piecewise function. The ability to decipher and represent these functions mathematically is a valuable skill.

• Definition: A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the input (x) values.

• Representation: They are typically written in a format that shows each sub-function and its corresponding interval:

  f(x) = {
      expression_1,  if condition_1
      expression_2,  if condition_2
      ...
  }
  

• Key Components:

  • Sub-functions: These are the individual functions (linear, quadratic, constant, etc.) that make up the piecewise function.
  • Intervals/Domains: Each sub-function is defined over a specific interval of x-values. These intervals must not overlap to ensure the function is well-defined (i.e., for any x, there's only one output f(x)).
  • Endpoints and Open/Closed Intervals: Pay close attention to how the intervals are defined at their endpoints. A closed interval (using brackets [ ]) includes the endpoint, while an open interval (using parentheses ( )) excludes it. This is crucial for determining where the function is continuous or discontinuous. Open circles on a graph usually represent open intervals, and closed circles represent closed intervals.

Steps to Determine the Equation from a Graph

Here's a systematic approach to find the equation of a piecewise function when you have its graph:

1. Identify the Intervals: Look at the x-axis and identify the different intervals where the function's behavior changes. These are the points where the "pieces" connect (or disconnect). Note the x-values at these points.

2. Determine the Type of Function in Each Interval: Examine each piece of the graph within its interval. Is it a straight line (linear)? A curve (quadratic, cubic, etc.)? A horizontal line (constant)?

3. Find the Equation of Each Piece:

   • Linear Functions: Use the slope-intercept form (y = mx + b) or the point-slope form (y - y1 = m(x - x1)). Find the slope (m) using two points on the line, and the y-intercept (b) where the line crosses the y-axis. If you have a point and the slope, the point-slope form is very helpful.
   • Constant Functions: These are horizontal lines, so their equation is simply y = c, where c is the y-value of the line.
   • Quadratic Functions: Look for the vertex and use the vertex form y = a(x - h)^2 + k, where (h, k) is the vertex. Alternatively, if you know the roots (x-intercepts), you can use the factored form y = a(x - r1)(x - r2). You'll need at least one other point on the parabola to solve for the leading coefficient, a.
   • Other Functions: For more complex functions, you might need to recognize the shape and use transformations of basic functions (e.g., y = sqrt(x), y = |x|, y = 1/x).

4. Determine the Domain for Each Piece: Write down the interval of x-values for which each piece of the function is defined. Use inequalities to express these intervals. Pay very close attention to whether the endpoints are included (≤ or ≥) or excluded (< or >). This is determined by open or closed circles on the graph at the endpoints.

5. Write the Piecewise Function: Combine the equations you found in step 3 and the domains you found in step 4 into the standard piecewise function notation.

Illustrative Examples

Let's walk through a few examples to solidify the process.

Example 1: A Simple Piecewise Function

Imagine a graph with two pieces:

• Piece 1: A straight line from x = -2 to x = 1 (inclusive), passing through the points (-2, 1) and (1, 4).
• Piece 2: A horizontal line from x = 1 (exclusive) to x = 3 (inclusive) at y = 5.

Here's how to find the equation:

1. Intervals: x = -2 to x = 1, and x = 1 to x = 3.

2. Function Types: Piece 1 is linear, Piece 2 is constant.

3. Equations:

   • Piece 1 (Linear):
     • Slope: m = (4 - 1) / (1 - (-2)) = 3/3 = 1
     • Using point-slope form with the point (-2, 1): y - 1 = 1(x - (-2)) => y - 1 = x + 2 => y = x + 3
   • Piece 2 (Constant): y = 5

4. Domains:

   • Piece 1: -2 ≤ x ≤ 1 (closed interval because the point (1, 4) lies on the line)
   • Piece 2: 1 < x ≤ 3 (open at x = 1 because Piece 1 already defines the function at x = 1)

5. Piecewise Function:

   f(x) = {
       x + 3,  if -2 ≤ x ≤ 1
       5,      if 1 < x ≤ 3
   }
   

Example 2: A Piecewise Function with a Quadratic

Suppose we have a graph with two pieces:

• Piece 1: A parabola from x = -3 (inclusive) to x = 0 (inclusive), with a vertex at (-1, -2) and passing through the point (0, -1).
• Piece 2: A straight line from x = 0 (exclusive) to x = 2 (inclusive), passing through (0, 1) and (2, 5).

Let's find the equation:

1. Intervals: x = -3 to x = 0, and x = 0 to x = 2.

2. Function Types: Piece 1 is quadratic, Piece 2 is linear.

3. Equations:

   • Piece 1 (Quadratic):
     • Using vertex form y = a(x - h)^2 + k with vertex (-1, -2): y = a(x + 1)^2 - 2
     • Substitute the point (0, -1): -1 = a(0 + 1)^2 - 2 => -1 = a - 2 => a = 1
     • So, y = (x + 1)^2 - 2 = x^2 + 2x - 1
   • Piece 2 (Linear):
     • Slope: m = (5 - 1) / (2 - 0) = 4/2 = 2
     • Y-intercept: The line passes through (0, 1), so b = 1
     • Therefore, y = 2x + 1

4. Domains:

   • Piece 1: -3 ≤ x ≤ 0 (closed interval)
   • Piece 2: 0 < x ≤ 2 (open at x = 0 because Piece 1 already defines the function there)

5. Piecewise Function:

   f(x) = {
       x^2 + 2x - 1,  if -3 ≤ x ≤ 0
       2x + 1,        if 0 < x ≤ 2
   }
   

Key Considerations and Potential Pitfalls

• Continuity: A piecewise function can be continuous or discontinuous at the points where the pieces connect. If the values of the two sub-functions are equal at the connecting point, the function is continuous there. Otherwise, it is discontinuous. Visually, a continuous function can be drawn without lifting your pen, while a discontinuous function has "jumps" or "breaks."

• Endpoints: Carefully determine whether the endpoints of the intervals are included or excluded based on the graph. This affects the inequalities used in the domain. Misidentifying an open vs. closed interval is a common error.

• Algebraic Manipulation: Be comfortable with algebraic manipulation to find the equations of the sub-functions. Practice with different forms of linear and quadratic equations.

• Special Cases: Be aware of special cases like vertical lines (x = c), which are not functions (they fail the vertical line test), and absolute value functions, which can often be expressed as piecewise functions. Remember that the absolute value of x is defined as:

  |x| = {
      x,  if x ≥ 0
     -x,  if x < 0
  }
  

• Transformations: Recognize how transformations (shifts, stretches, reflections) affect the basic functions. This can help you quickly determine the equation of a piece. For example, y = (x - 2)^2 is a parabola shifted 2 units to the right compared to y = x^2.

Advanced Scenarios

Here are some slightly more challenging scenarios you might encounter:

• More than Two Pieces: The process is the same, but you'll have more sub-functions and intervals to consider.
• Non-Linear Functions: You might encounter cubic functions, square root functions, rational functions, or trigonometric functions. Knowing the basic shapes of these functions is essential.
• Functions Defined by Limits: In more advanced contexts, the pieces of a piecewise function might be defined using limits, especially when dealing with removable discontinuities.

Practical Applications

Piecewise functions are not just abstract mathematical concepts; they have numerous real-world applications:

• Tax Brackets: As mentioned earlier, tax systems often use piecewise functions to determine the tax rate based on income.
• Shipping Costs: Shipping costs might have different rates depending on the weight or size of the package.
• Utility Bills: Electricity or water bills can have tiered pricing, where the cost per unit changes based on consumption.
• Step Functions: Step functions (a type of piecewise function) are used in digital signal processing and control systems.
• Modeling Physical Phenomena: Piecewise functions can model situations with sudden changes, such as the force of friction changing when an object starts moving.

Exercises for Practice

To master finding equations of piecewise functions from graphs, practice is key. Here are some exercises you can try:

1. Sketch a graph of a piecewise function with three linear pieces. Then, find the equation of the function you graphed.
2. Find graphs online of piecewise functions and try to determine their equations.
3. Create your own piecewise functions with different types of sub-functions (linear, quadratic, constant) and graph them.

Conclusion

Determining the equation of a piecewise function from its graph requires careful observation, algebraic skill, and attention to detail. By systematically identifying the intervals, function types, equations, and domains, you can successfully represent these functions mathematically. Understanding piecewise functions is a valuable skill with applications in various fields, making it a worthwhile topic to master. Always remember to double-check your work and pay close attention to endpoints and continuity!