Problem 5 - Algebra 2

Question

Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms. $$ y=(x-2)(x+5) $$

Step-by-Step Solution

Answer

Expanding \( (x - 2)(x + 5) \) results in \( x^2 + 3x - 10 \) with \(x^2 \) as the quadratic term, \(3x \) as the linear term, and \( -10 \) as the constant term. Hence, the function is quadratic.

Step 1: Expand the given expression

To determine the nature of the function, expand the bi- nomial multiplication \( (x - 2)(x + 5) \) using the distributive property (FOIL method).

Step 2: Identify the terms

After expanding, identify the type of each term in the expression. A quadratic term has the form \( ax^2 \) where \( a \) is a nonzero constant, a linear term has the form \( bx \) where \( b \) is a constant, and a constant term is a standalone number.

Step 3: Determine the type of function

After identifying the terms, determine if the function contains a quadratic term (\(x^2 \) term). If it does, and it's the term with the highest power, the function is quadratic.

Key Concepts

Linear Term

In the world of algebra, understanding the role of terms in a polynomial is vital. A linear term is one of the cornerstone pieces in this mathematical puzzle. It can be recognized by its simple form, \( bx \), where \( b \) is any real number, known as the coefficient, and \( x \) is the variable raised to the power of one, although this power is usually not written out. For instance, in the function \( 2x+3 \), \( 2x \) is the linear term.

The linear term plays a pivotal role in determining the graph of the equation. It influences the slope of the line, which is the rate at which the function increases or decreases. A function that only contains a linear term and a constant term will graph as a straight line and hence it’s called linear. If you include a linear term in a quadratic function, it will not alter the fact that the function is quadratic; it simply helps to dictate the shape of the parabola.

Constant Term

Moving on to the constant term, you'll find that this is the simplest type of term in a polynomial. It's a term that does not contain any variable, hence it remains constant. Its form is simply \( c \), where \( c \) is a real number. Let's consider the function \( 4x^2 - 2x + 7 \), here \( 7 \) is our constant term.

The constant term has a special job. In the graph of a linear function, it represents the y-intercept, the point where the line crosses the y-axis. In the case of a quadratic function, the constant term can affect the vertical position of the parabola, influencing where it crosses the y-axis. It's important to keep in mind that the presence of a constant term doesn't determine whether a function is linear or quadratic—a constant term can appear in both types of functions.

FOIL Method

When it comes to expanding binomials, the FOIL method comes to the rescue. FOIL stands for First, Outside, Inside, Last and it is a shortcut for remembering the steps required to multiply two binomials. To apply this method, let's breakdown the multiplication of the given binomials \( (x-2)(x+5) \) in our exercise.

First, we multiply the first terms from each binomial, \( x \) and \( x \), giving us \( x^2 \). Next, we multiply the outside terms, \( x \) and \( +5 \), resulting in \( +5x \). Then we move onto the inside terms, \( -2 \) and \( x \), to get \( -2x \). Lastly, we multiply the last terms, \( -2 \) and \( +5 \), yielding \( -10 \). Now, combining these results, we get \( x^2 + 3x - 10 \), which constitutes our expanded quadratic function.

The FOIL method is a fundamental skill for algebra students as it is used not only for expanding expressions but also for factoring, simplifying expressions, and solving polynomial equations.