IM Alg1.7.23 Lesson: Using Quadratic Expressions in Vertex Form to Solve Problems

Author:GeoGebra Classroom Activities, IM Algebra 1, Geometry, Algebra 2

Here are graphs that represent two functions, f and g, defined by:

can be expressed in words as “the value of

when

is 1.” Find or compute: the value of

when

is 1

Can you find an value that would make : Less than 1?

Greater than 10,000?

can be expressed in words as “the value of when is 9.” Find or compute: the value of when is 9

Can you find an value that would make : Greater than 7?

Less than -10,000?

The graph that represents has its vertex at . Here is one way to show, without graphing, that corresponds to the minimum value of .

When , the value of is 0, because .
Squaring any number always results in a positive number, so when is any value other than 8, will be a number other than 0, and when squared, will be positive.
Any positive number is greater than 0, so when , the value of will be greater than when . In other words, has the least value when .

Use similar reasoning to explain why the point

corresponds to the maximum value of

, defined by

Here are some quadratic functions, and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know.

Here is a portion of the graph of function , defined by .

is a rectangle. Points

and

coincide with the

-intercepts of the graph, and segment

just touches the vertex of the graph.

Find the area of

. Show your reasoning.

A function , defined by , describes the revenue collected from the sales of tickets for Performance A, a musical. The graph represents a function that models the revenue collected from the sales of tickets for Performance B, a Shakespearean comedy. In both functions, represents the price of one ticket, and both revenues and prices are measured in dollars. Without creating a graph of , determine which performance gives the greater maximum revenue when tickets are dollars each. Explain or show your reasoning.

IM Alg1.7.23 Lesson: Using Quadratic Expressions in Vertex Form to Solve Problems

Here are graphs that represent two functions, f and g, defined by:

Here are some quadratic functions, and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know.

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