Problem 11 - Algebra 2

Question

State whether each equation or function is linear. Write yes or no. If no, explain your reasoning. \(f(x)=6 x-19\)

Step-by-Step Solution

Answer

Yes, it's a linear equation.

Step 1: Identify the structure of the equation

Firstly, take a look at the function given, which is \( f(x) = 6x - 19 \). Notice how the function is presented in terms of \( x \).

Step 2: Recognize the form of a linear function

Recall that a linear function can be represented in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. This form is known as the slope-intercept form of a linear equation.

Step 3: Compare the given function with the linear form

Compare \( f(x) = 6x - 19 \) with the standard linear form \( ax + b \). Here, \( a = 6 \) and \( b = -19 \), and there are no exponents or non-linear terms such as \( x^2 \), \( \sqrt{x} \), etc.

Step 4: Determine if the function is linear

Since \( f(x) = 6x - 19 \) matches the linear form \( ax + b \), where both \( a = 6 \) and \( b = -19 \) are constants, the equation represents a straight line when graphed.

Key Concepts

Function Analysis

When analyzing a function, one must understand its structure and the elements it consists of. Consider the given function \( f(x) = 6x - 19 \). Here, we need to determine whether it's a linear equation. A linear function is characterized by its ability to create a straight line graph when plotted. In the context of function analysis:

• The components \( 6x \) and \(-19\) are crucial in determining the line's orientation and position.
• For an equation to be linear, the exponent of \( x \) must be one, meaning no quadratic or higher degree terms, radicals, or other non-linear transformations should be present.
• Therefore, analyzing function \( f(x) = 6x - 19 \) shows us it consists of a first-degree term \( 6x \) and a constant \(-19\), confirming it as linear.

By understanding these components, one can efficiently decide if the function belongs to the family of linear relations.

Slope-Intercept Form

The slope-intercept form of a linear equation is a standard format that makes identifying the characteristics of lines straightforward. It is expressed as \( f(x) = ax + b \). Here, the key components are:

• \( a \) is the slope of the line: It dictates the steepness and direction.
• \( b \) is the y-intercept: The point where the line crosses the y-axis.

In the problem, the function \( f(x) = 6x - 19 \) aligns with the slope-intercept form:

• The slope \( a = 6 \) suggests a line rising steeply, moving one unit to the right increases the y-value by 6.
• The y-intercept \( b = -19 \) tells us the line crosses the y-axis at -19.

Recognizing this form helps in promptly graphing lines and understanding the relationship of variables within a linear context.

Constants in Equations

In the context of linear equations like \( f(x) = 6x - 19 \), recognizing constants is crucial. Constants are fixed values that don’t change with \( x \). Their role:

• \( b \) (here, \(-19\)) decides the vertical position of the line on the graph.
• In general, constants define where the line will start when plotting on a grid.

The constant \( -19 \) implies that if \( x = 0 \), \( f(x) \) will equal \( -19 \), thus showing the line’s baseline position.
Considering both constants and coefficients provides insights into the equation's graph and helps to visualize where and how a line exists on the coordinate plane.