Graphing Quadratic Functions From the Standard Form

Videos and solutions to help Algebra I students learn how to graph a variety of quadratic functions using the form ax² + bx + c = 0 (standard form)

Students analyze and draw conclusions about contextual applications using the key features of a function and and its graph.

New York State Common Core Math Module 4, Algebra I, Lesson 17

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Lesson 17 Summary

The standard form of a quadratic function is f(x) = ax² + bx + c, where a ≠ 0. A general strategy to graphing a quadratic function from the standard form is:

• Look for hints in the function’s equation for general shape, direction, and y-intercept.
• Solve f(x) = 0 to find the x-intercepts by factoring, completing the square, or using the quadratic formula.
• Find the vertex by completing the square or using symmetry. Find the axis of symmetry and the x-coordinate of the vertex using -b/(2a) and the y-coordinate of the vertex by finding f(-b/(2a))
• Plot the points that you know (at least three are required for a unique quadratic function), sketch the graph of the curve that connects them, and identify the key features of the graph.

1. Graph the equation f(x) = x² - 5x + 6 and identify the key features.

2. Graph the equation f(x) = 1/2x² + 5x + 6 and identify the key features.

3. Paige wants to start a summer lawn-mowing business. She comes up with the following profit function that relates the total profit to the rate she charges for a lawn-mowing job:
P(x) = -x2 + 40x = 100

Both profit and her rate are measured in dollars.
a. Graph the function in order to answer the following questions.

b. According to the function, what is her initial cost (e.g., maintaining the mower, buying gas, advertising)? Explain your answer in the context of this problem.

c. Between what two prices does she have to charge to make a profit?

d. If she wants to make a profit this summer is this the right business choice?