Problem 61 - Algebra 1: Concepts and Skills

Question

Graph the function. (Lesson 4.8) $$ f(x)=-3 x+4 $$

Step-by-Step Solution

Answer

The graph of the equation \(f(x) = -3x + 4\) is a straight line with a slope of -3 and a y-intercept of 4. The line moves down and to the right, passing through the point (0, 4).

Step 1: Identify Slope and Y-intercept

The standard form for a linear equation is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Here, \(m = -3\) and \(b = 4\). The slope, -3, means that for every step forward on the x-axis, the function moves three steps downwards. The y-intercept, 4, is the point where the line crosses the y-axis.

Step 2: Plot the Y-intercept

Start by putting a point at the y-intercept, which is where \(y = 4\) on the y-axis. This is the starting point for the line.

Step 3: Use the Slope to Determine Additional Points

From the y-intercept, use the slope to identify more points on the line. The slope is step/run, which is -3/1 in this case. This means to move 1 step to the right and 3 steps down. Put a point at each of these locations and continue this process to generate several points.

Step 4: Draw the line

Once several points have been plotted, connect them with a straight line, making sure the line extends beyond the furthest points on either end.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a tool for easily graphing a straight line. The formula is written as \( y = mx + b \). This equation provides two valuable pieces of information:

• \( m \), the slope, indicates how steep the line is and in which direction it slopes.
• \( b \), the y-intercept, tells you where the line intersects the y-axis.

In the example equation \( f(x) = -3x + 4 \), you can directly see that the slope \( m \) is \(-3\) and the y-intercept \( b \) is 4. By understanding the slope-intercept form, you can quickly evaluate and predict the behavior of any linear function. You can instantly recognize how a change in either \( m \) or \( b \) will impact the line's angle and position on the graph.

Y-Intercept

The y-intercept is a crucial concept in graphing linear functions. This is the point where the line crosses the y-axis. For any function in the form \( y = mx + b \), the y-intercept is the value of \( y \) when \( x = 0 \). Simply put, it is \( b \) in the equation.
In our exercise, the y-intercept is 4. Thus, the line will cross the y-axis at the point \((0, 4)\).
Plotting the y-intercept is often the first step when graphing. It gives you a starting point to apply the slope and determine the line's direction across the graph. Being able to quickly identify the y-intercept helps in sketching the line more efficiently.

Slope of a Line

The slope of a line defines its steepness and direction. It is calculated as "rise over run," which means how much the line goes up or down for every step it moves horizontally.

• A positive slope rises rightward; the line ascends as you move from left to right.
• A negative slope descends rightward; the line declines as you move from left to right.

In our specific example, the slope is \(-3\). This negative slope indicates that as you move 1 unit to the right on the x-axis, the line goes down 3 units on the y-axis. By applying the slope repeatedly from the y-intercept, you can place multiple points along the line. These points help you draw the line accurately, depicting the graph of the function. Understanding the concept of slope is essential for predicting and creating the graphical representation of any linear function.