Problem 32 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=-30 x+20 $$

Step-by-Step Solution

Answer

Slope is -30; y-intercept is 20. Draw line through (0, 20) and (1, -10).

Step 1: Identifying the Slope and Y-intercept

The linear function is given in the form \( f(x) = -30x + 20 \). In a linear equation of the form \( y = mx + b \), \( m \) represents the slope and \( b \) represents the y-intercept. Here, the slope \( m = -30 \) and the y-intercept \( b = 20 \).

Step 2: Plotting the Y-intercept

Start by plotting the y-intercept on the graph. Since the y-intercept is 20, place a point on the y-axis at \( (0, 20) \).

Step 3: Using the Slope to Find Another Point

The slope \( m = -30 \) means that for every increase of 1 unit in \( x \), the value of \( y \) decreases by 30 units. From the y-intercept point \( (0, 20) \), move 1 unit to the right along the x-axis and then 30 units down to find a second point on the line, which is \( (1, -10) \).

Step 4: Drawing the Line

Using the two points \( (0, 20) \) and \( (1, -10) \), draw a straight line through these points. This line represents the graph of the function \( f(x) = -30x + 20 \).

Key Concepts

Slope

The slope is a key element of a linear function and tells us how steep the line is. In the equation of a line, written as \( y = mx + b \), the slope is represented by the letter \( m \). The slope can be thought of as a ratio or a fraction that shows the rate of change between the \( y \)-values and the \( x \)-values.

Here's what the slope does:

• It indicates the direction of the line. A positive slope will slant upwards from left to right, while a negative slope will slant downwards.
• The absolute value of the slope tells us how quickly the line rises or falls. A larger absolute value means a steeper line.

For the exercise you're working on, the slope \( m \) is \(-30\). This means that as \( x \) increases by one unit, \( y \) decreases by 30 units. So, the line is quite steep, and it slants downwards. Understanding the slope helps predict how changes in \( x \) affect changes in \( y \).

Y-intercept

The y-intercept is another fundamental aspect of linear functions and is found in the equation \( y = mx + b \) as the term \( b \). The y-intercept is the point where the line crosses the y-axis, which is the line where \( x = 0 \).

Properties of the y-intercept include:

• It is always found on the y-axis, represented by the coordinates \( (0, b) \).
• This point is crucial for graphing as it provides a starting point to draw the line.

In your exercise, the y-intercept \( b \) is 20, meaning the line intersects the y-axis at the point \( (0, 20) \). This gives you a point exactly where the line hits the y-axis and is essential in plotting the initial point on the graph before using the slope to find other points.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points. The equation \( y = mx + b \) provides all the information needed for this: the slope \( m \) and the y-intercept \( b \).

Here's a simple way to graph a linear equation:

1. Start by plotting the y-intercept \( (0, b) \) on the y-axis.
2. Use the slope \( m \) as a guide to find another point. The slope tells you how many units to move up (or down) and to the right (or left). For example, with a slope of \(-30\), move 1 unit right and 30 units down from the y-intercept to find the next point \( (1, -10) \).
3. Draw a straight line through these points to reveal the graph of the linear equation.

Graphing helps visually represent the relationship between variables. By following simple steps of plotting the y-intercept and utilizing the slope, you can easily draw any linear equation on a graph.