Problem 57 - Algebra 2

Question

Find the slope and the intercepts of each line. $$ g(x)=54 x-1 $$

Step-by-Step Solution

Answer

The slope of the line is 54, the y-intercept is at the point \((0, -1)\), and the x-intercept is at the point \((\frac{1}{54}, 0)\).

Step 1: Identify the Slope

Given the linear function in slope-intercept form, which is defined as \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept, we can directly compare it to \( g(x) = 54x - 1 \). Here, the coefficient of \( x \), which is 54, is the slope of the line.

Step 2: Find the y-intercept

The y-intercept is the constant term in the linear function when it's written in slope-intercept form. For the equation \( g(x) = 54x - 1 \), the y-intercept is \( -1 \), which means the line crosses the y-axis at the point \( (0, -1) \).

Step 3: Find the x-intercept

To find the x-intercept, set the output of the function \( g(x) \) to zero and solve for \( x \). The equation becomes \( 0 = 54x - 1 \). Adding 1 to both sides gives us \( 1 = 54x \). Dividing both sides by 54 gives us \( x = \frac{1}{54} \). Therefore, the line crosses the x-axis at the point \( (\frac{1}{54}, 0) \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is the go-to equation for quickly graphing linear functions. It's represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form allows you to see at a glance how steep the line is and where it crosses the y-axis. For example, in the given function \( g(x) = 54x - 1 \), we can identify \( m = 54 \) as the slope and \( b = -1 \) as the y-intercept. Understanding this format makes plotting lines a straightforward process.

Linear Equations

Linear equations are the simplest type of equations used for lines in algebra. They have one or two variables with no exponents more significant than one, resulting in a straight line when graphed. Our example \( g(x) = 54x - 1 \) is a linear equation with a single variable. The solutions to these equations are ordered pairs that, when connected, form a line on a graph. Recognizing the structure of linear equations will aid in anticipating the shape and direction of the line.

Y-Intercept

The y-intercept is where a line crosses the y-axis. In slope-intercept form, it's denoted by \( b \). This value tells us the exact point on the y-axis where the line will be, regardless of the rest of its course. For \( g(x) = 54x - 1 \), \( b = -1 \) indicates that our line will intersect the y-axis at \( (0, -1) \). This point is a crucial starting mark for graphing and understanding the line's position relative to the origin.

X-Intercept

In contrast, the x-intercept is where the line crosses the x-axis. To find this point, we set the output of the function to zero and solve for \( x \). Using the given line \( g(x) = 54x - 1 \) as an example, when we make \( g(x) = 0 \) and solve the resulting equation, we discover the x-intercept to be \( (\frac{1}{54}, 0) \). This intercept is another anchor point that helps draft the graph of the line and marks where the line will cross the x-axis.

Graphing Linear Functions

Graphing linear functions like \( g(x) = 54x - 1 \) begins with plotting the y-intercept. From there, we use the slope, or the 'rise over run,' to determine the direction and steepness of the line. A steeper slope means a greater vertical change for each unit of horizontal change. For every unit you move horizontally from the y-intercept, the slope tells you how many units to move up or down to find the next point. Drawing a line through these points will give you the graph of the linear function. Always remember to check for both intercepts, as they confirm that your graph is accurate.