Problem 46 - Precalculus Essentials

Question

Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$y=-\frac{2}{5} x+6$$

Step-by-Step Solution

Answer

The slope of the function \(y=-\frac{2}{5}x+6\) is -\frac{2}{5} and the y-intercept is 6.

Step 1: Identify the slope

The slope (\(m\)) is the coefficient of \(x\) in the given equation. So for the equation \(y=-\frac{2}{5}x+6\), the slope is equal to -\frac{2}{5}.

Step 2: Identify the y-intercept

The y-intercept (\(b\)) is the constant in the equation. So for the equation \(y=-\frac{2}{5}x+6\), the y-intercept is equal to 6

Step 3: Plot the y-intercept

Begin by plotting the y-intercept on the graph at the point where \(y=6\). This is the point where the line crosses the y-axis.

Step 4: Use the slope to plot the next point

The slope -\frac{2}{5} means that for each movement of 5 units to the right on the x-axis, we move 2 units down on the y-axis. From the y-intercept point, move 5 units to right and 2 units down to determine the next point that the line goes through.

Step 5: Draw the line

Draw a straight line that passes through the y-intercept and the point determined by the slope. This line represents the graph of the function \(y=-\frac{2}{5}x+6\).

Key Concepts

Understanding Slope

When talking about linear equations, the slope is a crucial component to understand. The slope, often represented by the letter \( m \), tells us how steep a line is. It indicates the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on a line. In simple terms, it shows how much the y-value increases or decreases as the x-value increases by one unit.

• If the slope is positive, it means the line rises as you move from left to right.
• If the slope is negative, the line falls as you move from left to right.
• A larger absolute value of the slope indicates a steeper line.
• If the slope is zero, the line is perfectly horizontal.

For example, in the equation \( y = -\frac{2}{5}x + 6 \), the slope is \( -\frac{2}{5} \). This negative value means that the line falls as we move from left to right. Specifically, for every 5 units you move right on the x-axis, you move down 2 units on the y-axis.

Defining Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is an essential concept in graphing because it provides a starting point for plotting a linear function. In the slope-intercept form of a linear equation, \( y = mx + b \), the \( b \) represents the y-intercept. This means when \( x = 0 \), \( y \) will be equal to \( b \).

• The y-intercept is always located at the point \((0, b)\) on the graph.
• If a line crosses the y-axis higher up, it will have a larger y-intercept value.
• If it crosses lower down, it will have a smaller y-intercept value.

In the equation \( y = -\frac{2}{5}x + 6 \), the y-intercept is 6. This tells us that the line will intersect the y-axis at the point \((0, 6)\). Knowing the y-intercept is the first step in plotting the graph.

Graphing Linear Functions

Graphing a linear function involves using the slope and y-intercept we have identified to create a visual representation on a coordinate plane. The process can be straightforward once you understand the relationship between these two elements.

To graph a linear equation such as \( y = -\frac{2}{5}x + 6 \):

• Start by plotting the y-intercept. This is where the line crosses the y-axis, so you would place a point at the coordinate \((0, 6)\).
• Next, use the slope to find another point on the line. From the y-intercept, move 5 units to the right (because of the slope denominator) and 2 units down (because of the slope numerator being negative).
• Plot this new point. It provides a second point the line must pass through.
• For extra accuracy, you can continue moving along the slope to plot more points.
• Finally, draw a straight line through these two points. Extend it across the graph to show that it continues indefinitely in both directions.

This visualization helps in understanding how changes in the slope and y-intercept affect the graph of a linear equation.