Analyzing a Graph

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Common Core For Algebra I

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New York State Common Core Math Algebra I, Module 1, Lesson 1

Worksheets for Algebra I, Module 1, Lesson 1 (pdf)

Student Outcomes

  • From a graphic representation, students recognize the function type, interpret key features of the graph, and create an equation or table to use as a model of the context for functions addressed in previous modules (i.e., linear, exponential, quadratic, cubic, square root, cube root, absolute value, and other piecewise functions).

Analyzing a Graph


Opening Exercise

The graphs below give examples for each parent function we have studied this year. For each graph, identify the function type and the general form of the parent function’s equation; then offer general observations on the key features of the graph that helped you identify the function type. (Function types include linear, quadratic, exponential, square root, cube root, cubic, absolute value, and other piecewise functions. Key features may include the overall shape of the graph, 𝑥𝑥- and 𝑦𝑦-intercepts, symmetry, a vertex, end behavior, domain and range values or restrictions, and average rates of change over an interval.)

Example 1

Eduardo has a summer job that pays him a certain rate for the first 40 hours each week and time-and-a-half for any overtime hours. The graph below shows how much money he earns as a function of the hours he works in one week.


  1. Write the function in analytical (symbolic) form for the graph in Example 1.
    a. What is the equation for the first piece of the graph?
    b. What is the equation for the second piece of the graph?
    c. What are the domain restrictions for the context?
    d. Explain the domain in the context of the problem.

For each graph below use the questions and identified ordered pairs to help you formulate an equation to represent it.
2. Function type:
Parent function:

Lesson Summary

  • When given a context represented graphically, you must first:
  • Identify the variables in the problem (dependent and independent), and
  • Identify the relationship between the variables that are described in the graph or situation.
  • To come up with a modeling expression from a graph, you must recognize the type of function the graph represents, observe key features of the graph (including restrictions on the domain), identify the quantities and units involved, and create an equation to analyze the graphed function.
  • Identifying a parent function and thinking of the transformation of the parent function to the graph of the function can help with creating the analytical representation of the function.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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