2.2+Changing+One+Dimension



J an. 27, 2009 AvS Big Idea: Quadratic functions can help us describe situations in the real world. Essential Question #2: What does a quadratic function look like in a situation, table, graph, or equation? =Problem 2.2: Changing One Dimension =

Notes from Class:

 * A = n^2 + 2n
 * A = n(n+2)
 * An expression uses mathematical operations to combine numbers and/or variables (i.e. 7 + 3⁴). An expression can be simplified.
 * An equation shows two equal expressions, usually written by symbols that are separated into two different sides which are joined by an equal sign. An equation can be solved.
 * Expanded form: An expression that is written as the sum or difference of terms (also //term form//) *addition
 * Factored form: An expression that is written as the product of two linear factors. *multiplication
 * Quadratic equation: y = ax^2 + bx + c

Problem 2.2
The dimensions of the added rectangle are 2 by x. The area is 2x. A = x^2 + 2x The new rectangles length is x and the width is 2. A = 2x. A = x^2 + 2x. The first graph is a parabola. It shows a quadratic equation A = 2x. The second graph is a straight line. It shows a linear equation.
 * A) **** A square has sides of length x centimeters. A new rectangle is created by increasing one dimension of the square by 2 centimeters. **
 * 1) ** The new rectangle is made up of the original square and an added rectangle. What are the dimensions of the added rectangle? What is its area? **
 * 1) ** Write an equation for the area of the new rectangle as the sum of the area of the original square and the area of the added rectangle. **
 * 1) ** What are the length and the width of the new rectangle? Write an equation for the area of the new rectangle as its length times its width. **
 * 1) ** Graph your equations from parts 2 and 3. Describe the shapes of the graphs. How do the graphs compare? What does this tell you about the two equations? **

The new rectangles area is 3x. The squares is x^2. A = x^2 + 3x A = x (x + 3) A = x^2 + 3x. The first graph is a parabola curving downward. A = x (x + 3). The second graph is also a parabola curving downwards. Both graphs represent quadratic functions. This graph is exactly the same as the other graph.
 * B) **** A square has sides of length x centimeters. One dimension of the square is increased by 3 centimeters to create a new rectangle. **
 * 1) ** How much greater is the area of the new rectangle than the area of the square? **
 * 1) ** Write two equations for the area of the new rectangle. **
 * 1) ** Graph both equations on your calculator. Describe the shapes of the graphs. How do the graphs compare? **

Problem 2.2 Follow-Up
A = x (x + 4) >> factored form A = x^2 + 4x >> expanded form A = x (x – 6) >> factored form A = x^2 – 6x >> expanded form A = x^2 + 4x A = x^2 – 4x A = 5x + 2x A = x (x + 5) A = x (x – 5) A = x (5 + 4) If it is written in expanded form, you can write it as an equivalent expression in factored form to see if it is a quadratic equation.
 * 1) The diagram below shows a rectangle divided into two smaller rectangles. Write two expressions, one in factored form and one in expanded form, for the area of the large rectangle. **
 * 2) Write two expressions, one in factored form and one in expanded form, for the area of the unshaded region. **
 * 3) Each expression below represents the area of a rectangle. Write an equivalent expression in expanded form. **
 * x (x + 4) x (x – 4) x (5 + 2) **
 * 4) Each expression below represents the area of a rectangle. Write an equivalent expression in factored form. **
 * X^2 + 5x x^2 = 5x 5x + 4x **
 * 5) An equation that describes a quadratic function is called a //quadratic equation.// If one side of an equation is not written in factored form, how can you tell whether it is a quadratic equation? **