Problem 84 - Algebra A Combined Function

Question

Graph each linear function. \(f(x)=-4 x\)

Step-by-Step Solution

Answer

The linear function \( f(x) = -4x \) is a straight line through the origin with a slope of -4.

Step 1 - Identify the Linear Equation

The given function is a linear equation in the form of \( f(x) = -4x \). This equation represents a straight line where the slope is \(-4\) and the y-intercept is \(0\).

Step 2 - Determine the Y-Intercept

The function \( f(x) = -4x \) can also be written in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( b = 0 \), so the line passes through the origin (0,0).

Step 3 - Use the Slope to Find Another Point

Since the slope \( m \) is \(-4\), it means that for every unit increase in \( x \), \( y \) decreases by 4 units. So, starting from the origin (0,0), if we move 1 unit to the right (increment \( x \) by 1), the \( y \) value will be \(-4 \times 1 = -4\), giving us the point (1, -4).

Step 4 - Plot the Points on a Graph

Plot the points (0,0) and (1,-4) on the Cartesian plane. Draw a straight line through these points which extends in both directions.

Step 5 - Draw the Graph

The straight line through (0,0) and (1,-4) is the graph of the function \( f(x) = -4x \). This line will slope downwards to the right, illustrating the negative slope.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra. It involves plotting points on a Cartesian plane and drawing a line through these points to represent a linear function. A linear equation, like the one given in the exercise \(f(x) = -4x\), is recognized by its format, where it defines a straight line.
When graphing a linear equation, it is crucial to identify key components such as the slope and the y-intercept. You start by choosing values for \(x\) and computing the corresponding \(y\) values using the given function.

• Plot the points on the graph.
• Draw a line connecting the plotted points.

The line you draw should extend in both directions, capturing the endless nature of a linear function on the coordinate plane.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most useful ways to express such equations. It is written as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) indicates the y-intercept. This form simplifies finding the slope and intercept, allowing you to quickly sketch a graph.
For example, in the function \(f(x) = -4x\), you can rewrite it as \(y = -4x + 0\).

• The slope \(m = -4\) shows that the line will descend as it moves from left to right.
• The y-intercept \(b = 0\) tells us the line crosses the y-axis at the origin.

By organizing a linear function in this way, it becomes straightforward to identify the slope and y-intercept, making graphing significantly easier.

Y-Intercept

The y-intercept is a critical component in graphing a linear function, indicating where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), \(b\) is the y-intercept.
In our example \(f(x) = -4x\), the y-intercept is \(0\), meaning the line starts at the origin \((0,0)\).

• To find the y-intercept, set \(x = 0\) and solve for \(y\).
• This point is one of the easiest to plot on the graph.

Understanding the y-intercept helps define the starting point for the graph, making it an essential step in visualizing the function.