Writing Transformed Piecewise Functions from Graphs
Writing transformed piecewise functions from graphs is a Grade 11 Algebra 1 skill from enVision Chapter 5 that reverse-engineers the piecewise definition from a graph. The 4-step process: identify domain intervals where rules change, determine the parent function type for each piece, identify transformations using key points, write in transformed form with domain restrictions. A graph showing |x - 2| + 1 for x <= 3 and -x + 7 for x > 3 becomes a piecewise function combining both pieces. Pay close attention to whether endpoints use open or closed circles when writing domain boundaries.
Key Concepts
To find a piecewise defined function from its graph, identify each piece and its transformations from the parent function. 1. Identify the domain intervals where different function rules apply. 2. For each piece, determine the parent function type (linear, quadratic, absolute value, etc.). 3. Identify the transformations applied to each parent function using key points from the graph. 4. Write each piece in transformed form, then combine into a piecewise function with appropriate domain restrictions.
Common Questions
How do you write a piecewise function from its graph?
Identify where the function rule changes (domain boundaries), determine the parent function for each piece, find the transformations from key points, and write each piece with its domain restriction.
What information do you need from each graph piece?
The domain interval (endpoints and whether they are included), the parent function type (linear, absolute value, quadratic), and key features like vertex, slope, or intercepts.
How do open and closed circles affect the piecewise definition?
A closed circle means the endpoint is included, using <= or >=. An open circle means excluded, using < or >.
A graph has |x-2|+1 for x<=3 and -x+7 for x>3. Write the piecewise function.
f(x) = |x-2|+1 if x<=3, and -x+7 if x>3. Note the boundary at x=3 is included in the first piece (closed circle at that end).
How do you identify that a graph piece is an absolute value transformation?
Look for a V-shape. The vertex gives the values of h and k in a|x-h|+k. Slope of each arm gives |a|.
What if the graph piece is a parabola?
Identify the vertex (h, k) and write f(x) = a(x-h)^2 + k. Use another point on the curve to solve for a.