Problem 7
Question
Determine whether each function is linear or quadratic. Identify the quadratic, linear, and constant terms. $$ h(x)=(3 x)(2 x)+6 $$
Step-by-Step Solution
Verified Answer
The function h(x) is quadratic with a quadratic term of 6x^2 and a constant term of 6.
1Step 1: Determine the Type of Function
Examine the given function by looking at the terms present in the expression. A linear function is of the form f(x) = ax + b, where a and b are constants. A quadratic function is typically of the form f(x) = ax^2 + bx + c, where a, b, and c are constants. To determine if the function is linear or quadratic, identify the highest power of the variable x.
2Step 2: Simplify the Function
Simplify the expression by multiplying out the terms. Multiply (3x) by (2x) to find the term that will reveal the highest power of x.
3Step 3: Identify the Terms
After simplification in the previous step, organize the terms to identify the quadratic, linear, and constant terms based on the powers of x.
Key Concepts
Identifying Quadratic TermsLinear vs Quadratic FunctionsSimplifying Algebraic Expressions
Identifying Quadratic Terms
Understanding the difference between linear and quadratic terms is key to analyzing functions. Quadratic terms are easily recognizable as they feature a variable raised to the second power – that is, squared. For instance, in a quadratic equation in the form of \( f(x) = ax^2 + bx + c \), the term \( ax^2 \) is the quadratic term.
To proficiently identify quadratic terms in an algebraic expression, look for exponents of 2 on the variable, such as in \( 3x^2 \), \( -7y^2 \), etc. Notably, if a function contains at least one term with a squared variable, it is typically a quadratic function. The function in our exercise, \( h(x) = (3x)(2x) + 6 \), contains the term \( (3x)(2x) \) which simplifies to \( 6x^2 \), clearly identifying it as a quadratic term and the function itself as quadratic.
To proficiently identify quadratic terms in an algebraic expression, look for exponents of 2 on the variable, such as in \( 3x^2 \), \( -7y^2 \), etc. Notably, if a function contains at least one term with a squared variable, it is typically a quadratic function. The function in our exercise, \( h(x) = (3x)(2x) + 6 \), contains the term \( (3x)(2x) \) which simplifies to \( 6x^2 \), clearly identifying it as a quadratic term and the function itself as quadratic.
Linear vs Quadratic Functions
Distinguishing between linear and quadratic functions is a fundamental skill in algebra. A linear function has the form \( f(x) = ax + b \), where the variable x is raised to the first power, and 'a' and 'b' are constants representing the slope and y-intercept, respectively.
On the other hand, a quadratic function typically takes the form \( f(x) = ax^2 + bx + c \), where 'a', 'b', and 'c' are constants, and the presence of an \( x^2 \) term indicates a parabolic curve when graphed. Unlike linear functions, which produce straight lines, quadratic functions generate curves that open upwards or downwards depending on the sign of the coefficient 'a'.
In our textbook exercise, the function \( h(x) \) is quadratic because it contains the product of two linear terms \((3x)\) and \((2x)\), which results in a quadratic term of \( 6x^2 \) after simplification.
On the other hand, a quadratic function typically takes the form \( f(x) = ax^2 + bx + c \), where 'a', 'b', and 'c' are constants, and the presence of an \( x^2 \) term indicates a parabolic curve when graphed. Unlike linear functions, which produce straight lines, quadratic functions generate curves that open upwards or downwards depending on the sign of the coefficient 'a'.
In our textbook exercise, the function \( h(x) \) is quadratic because it contains the product of two linear terms \((3x)\) and \((2x)\), which results in a quadratic term of \( 6x^2 \) after simplification.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a vital process to simplify the analysis of functions. Multiplying and combining like terms are common methods used to simplify. For the expression \( h(x) = (3x)(2x) + 6 \), we multiply to combine the terms:\
\( (3x)(2x) = 6x^2 \).
This multiplication demonstrates that the expression is a quadratic function. Moreover, there are no other terms involving x to be combined, thus the quadratic term in the function is \( 6x^2 \) and the constant term is 6. It's important to note that this function lacks a linear term, as all the variable terms are squared.
When simplifying, always perform operations in the correct order: first multiplication or division (whichever comes first from left to right), and then addition or subtraction (again, from left to right), combining like terms whenever possible. This process makes the function easier to understand, differentiate, and integrate, and is also critical when solving equations.
\( (3x)(2x) = 6x^2 \).
This multiplication demonstrates that the expression is a quadratic function. Moreover, there are no other terms involving x to be combined, thus the quadratic term in the function is \( 6x^2 \) and the constant term is 6. It's important to note that this function lacks a linear term, as all the variable terms are squared.
When simplifying, always perform operations in the correct order: first multiplication or division (whichever comes first from left to right), and then addition or subtraction (again, from left to right), combining like terms whenever possible. This process makes the function easier to understand, differentiate, and integrate, and is also critical when solving equations.
Other exercises in this chapter
Problem 7
Factor each expression. $$ x^{2}+3 x+2 $$
View solution Problem 7
Graph each function. Identify the axis of symmetry. $$ y=(x-5)^{2}-3 $$
View solution Problem 7
Graph each function. $$ y=\frac{1}{2} x^{2}+3 $$
View solution Problem 8
Solve each equation using the Quadratic Formula. $$ x^{2}+6 x-5=0 $$
View solution