Graphing Quadratic Equations From the Vertex Form

Examples, videos, and solutions to help Algebra I students learn how to graph simple quadratic equations of the form y = a(x-h)² + k (completed-square or vertex form), recognizing that (h, k) represents the vertex of the graph and use a graph to construct a quadratic equation in vertex form.

Students understand the relationship between the leading coefficient of a quadratic function and its concavity and slope and recognize that an infinite number of quadratic functions share the same vertex.

New York State Common Core Math Algebra I, Module 4, Lesson 16

Worksheets for Algebra 1

Lesson 16 Summary

When graphing a quadratic equation in vertex form, y= a(x-h)² + k, (h, k) are the coordinates of the vertex.

Opening Exercise
Graph the equations y = x², y = (x - 2)², and y = (x + 2)² on the interval -3 ≤ x ≤ 3.

Exercises 1–2

1. Without graphing, state the vertex for each of the following quadratic equations. a. y = (x - 5)² + 3
   b. y = x² - 2.5
   c. y = (x + 4)²
2. Write a quadratic equation whose graph will have the given vertex.
   a. (1.9, -4)
   b. (0, 100)
   c. (-2, 3/2)

Example 1
Caitlin has 60 feet of material that can be used to make a fence. Using this material, she wants to create a rectangular pen for her dogs to play in. What dimensions will maximize the area of the pen?
a. Let w be the width of the rectangular pen in feet. Write an expression that represents the length when the width is feet.
b. Define a function A(w) that describes the area, , in terms of the width, w.
c. Rewrite A(w) in vertex form.
d. What are the coordinates of the vertex? Interpret the vertex in terms of the problem.
e. What dimensions maximize the area of the pen? Do you think this is a surprising answer?

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