UNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction

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1 Prerequisite Skills This lesson requires the use of the following skills: translating verbal expressions to algebraic expressions evaluating expressions following the order of operations adding and subtracting polynomials Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know that a variable is a letter used to represent a value or unknown quantity that can change or vary. We have seen several linear expressions such as 2x + 1. In this example, the highest power of the variable x is the first power. In this lesson, we will look at expressions where the highest power of the variable is 2. Key Concepts A quadratic expression is an expression where the highest power of the variable is the second power. A quadratic expression can be written in the form ax 2 + bx + c, where x is the variable, and a, b, and c are constants. Both b and c can be any number, but a cannot be equal to 0 because quadratic expressions must contain a squared term. An example of a quadratic expression is 4x 2 + 6x 2. When a quadratic expression is set equal to 0, as in 4x 2 + 6x 2 = 0, the resulting equation is called a quadratic equation. A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where x is the variable, a, b, and c are constants, and a 0. The quadratic expression 4x 2 + 6x 2 is made up of many component parts: terms, factors, coefficients, and constants. A term is a number, a variable, or the product of a number and variable(s). There are 3 terms in the given expression: 4x 2, 6x, and 2. A factor is one of two or more numbers or expressions that when multiplied produce a given product. In the given expression, the factors of 4x 2 are 4 and x 2 and the factors of 6x are 6 and x. U5-6

2 The number multiplied by a variable in an algebraic expression is called a coefficient. In the given expression, the coefficient of the term 4x 2 is 4 and the coefficient of the term 6x is 6. When there is no number before a variable, the coefficient is either 1 or 1 because x means 1x and x 2 means 1x 2. A term that does not contain a variable is called a constant term because the value of the term does not change. In the given expression, 2 is a constant. Two or more terms that contain the same variables raised to the same power are called like terms. Like terms can be combined by adding. Be sure to follow the order of operations when combining like terms. 4x 2 + 6x 2 has no like terms, so let s use another expression as an example: 9x 2 8x 2 + 2x. In the expression 9x 2 8x 2 + 2x, 9x 2 and 8x 2 are like terms. After simplifying the expression by combining like terms (9x 2 and 8x 2 ), the result is x 2 + 2x. A monomial is a number, a variable, or the product of a number and variable(s). We can also think of a monomial as an expression containing only one term. 5x 2 is an example of a monomial. A polynomial is a monomial or the sum of monomials. A polynomial can have any number of terms. A binomial is a polynomial with two terms. 6x + 9 is an example of a binomial. A trinomial is a polynomial with three terms. 4x 2 + 6x 2 is an example of a trinomial. Common Errors/Misconceptions not following the order of operations incorrectly identifying like terms inaccurately combining terms involving subtraction incorrectly combining terms by changing exponents U5-7

3 Guided Practice Example 1 Identify each term, coefficient, and constant of 6(x 1) x(3 2x) Classify the expression as a monomial, binomial, or trinomial. Determine whether it is a quadratic expression. 1. Simplify the expression. The expression can be simplified by following the order of operations and combining like terms. 6(x 1) x(3 2x) + 12 Original expression 6x 6 x(3 2x) + 12 Distribute 6 over x 1. 6x 6 3x + 2x Distribute x over 3 2x. 3x x 2 Combine like terms: 6x and 3x; 6 and 12. 2x 2 + 3x + 6 Rearrange terms so the powers are in descending order. 2. Identify all terms. There are three terms in the expression: 2x 2, 3x, and Identify all coefficients. The number multiplied by a variable in the term 2x 2 is 2; the number multiplied by a variable in the term 3x is 3; therefore, the coefficients are 2 and Identify any constants. The quantity that does not change (is not multiplied by a variable) in the expression is 6; therefore, 6 is a constant. U5-8

4 5. Classify the expression as a monomial, binomial, or trinomial. The polynomial is a trinomial because it has three terms. 6. Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax 2 + bx + c, where a = 2, b = 3, and c = 6. Example 2 Translate the verbal expression take triple the difference of 12 and the square of x, then increase the result by the sum of 3 and x into an algebraic expression. Identify the terms, coefficients, and constants of the given expression. Is the expression quadratic? 1. Translate the expression by breaking it down into pieces. The difference of 12 and the square of x translates to (12 x 2 ). Triple this expression is 3(12 x 2 ). The sum of 3 and x translates to (3 + x). Increasing the original expression by this sum translates to 3(12 x 2 ) + (3 + x). 2. Simplify the expression. 3(12 x 2 ) + (3 + x) Expression 36 3x 2 + (3 + x) Distribute 3 over (12 x 2 ). 3x 2 + x + 39 Combine like terms. The simplified expression is 3x 2 + x Identify all terms. There are three terms in the expression: 3x 2, x, and 39. U5-9

5 4. Identify all coefficients. The number multiplied by a variable in the term 3x 2 is 3; the number multiplied by a variable in the term x is 1; therefore, 3 and 1 are coefficients. 5. Identify any constants. The number that is not multiplied by a variable in the expression is 39; therefore, 39 is a constant. 6. Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax 2 + bx + c, where a = 3, b = 1, and c = 39. Example 3 A fence surrounds a park in the shape of a pentagon. The side lengths of the park in feet are given by the expressions 2x 2, 3x + 1, 3x + 2, 4x, and 5x 3. Find an expression for the perimeter of the park. Identify the terms, coefficients, and constant in your expression. Is the expression quadratic? 1. Find an expression for the perimeter of the park. Add like terms to find the perimeter, P. P = 2x 2 + (3x + 1) + (3x + 2) + 4x + (5x 3) Set up the equation using the given expressions. P = 2x 2 + 3x + 3x + 4x + 5x P = 2x x The expression for the park s perimeter is 2x x. Reorder like terms. Combine like terms. 2. Identify all terms. There are two terms in this expression: 2x 2 and 15x. U5-10

6 3. Identify all coefficients. The number multiplied by a variable in the term 2x 2 is 2; the number multiplied by a variable in the term 15x is 15; therefore, 2 and 15 are coefficients. 4. Identify any constants. Every number in the expression is multiplied by a variable; therefore, there is no constant. 5. Determine whether the expression is a quadratic expression. It is a quadratic expression because it can be written in the form ax 2 + bx + c, where a = 2, b = 15, and c = 0. U5-11