Problem 29 - Algebra Form and Function

Question

Identify the slope and \(y\) -intercept and graph the function. $$ f(x)=-0.5 x-0.2 $$

Step-by-Step Solution

Answer

Answer: The slope of the given function is -0.5 and the y-intercept is -0.2. To graph the function, plot the y-intercept on the y-axis, then use the slope to find another point on the graph, and draw a straight line through the two points to represent the function \(f(x) = -0.5x - 0.2\).

Step 1: Identify the slope

The slope of the given function can be identified by comparing it to the standard slope-intercept form (y = mx + b). In this case, the slope (m) is -0.5. So, the slope of the function is: $$ m = -0.5 $$

Step 2: Identify the y-intercept

Similarly, the y-intercept (b) can be identified by comparing the given function to the standard slope-intercept form. In this case, the y-intercept (b) is -0.2. So, the y-intercept of the function is: $$ b = -0.2 $$

Step 3: Graph the function

Using the identified slope and y-intercept, we can now graph the linear function: 1. Begin by plotting the y-intercept on the y-axis. In this case, it is -0.2. 2. From the y-intercept, use the slope to find another point on the graph. Since the slope is -0.5, this means that for every one unit increase in x, y will decrease by 0.5 units. In this case, if we move 1 unit to the right from the y-intercept, we will move 0.5 units down as well. This gives us our second point. 3. Now that we have two points, we can draw a straight line through them. This line represents the graph of the function \(f(x) = -0.5x - 0.2\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear functions in a simple equation format. This format makes it very easy to read off the slope and y-intercept of the line. The formula for the slope-intercept form is given by:

• \( y = mx + b \)

In this equation:

• \( y \) represents the dependent variable, which usually changes based on the value of \( x \).
• \( m \) is the slope of the line, indicating the steepness and direction of the line.
• \( x \) is the independent variable, typically found along the horizontal axis.
• \( b \) is the y-intercept, which tells where the line crosses the y-axis.

Using this form, not only can we understand what the line looks like without graphing it, but also determine how it behaves as \( x \) changes. In the example provided, \( f(x) = -0.5x - 0.2 \), it's clear that the slope is \(-0.5\) and y-intercept is \(-0.2\). This clear structure is what makes the slope-intercept form a reliable tool for graphing linear functions.

Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form (\( y = mx + b \)), the slope is represented by \( m \). Slope essentially describes how much \( y \) changes as \( x \) changes. A positive slope means the line rises as you move from left to right, while a negative slope means the line falls.

• A slope of 0 implies the line is flat, parallel to the x-axis.
• The slope is calculated as the "rise" over the "run," or the change in \( y \) over the change in \( x \):\[ m = \frac{\Delta y}{\Delta x} \]

For the function \( f(x) = -0.5x - 0.2 \), the slope is \(-0.5\). This means:

• For every one unit increase in \( x \), \( y \) decreases by 0.5 units.
• The line will tilt downwards as you move from left to right, appearing to \"fall\" on the graph.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In mathematical terms, it's the value of \( y \) when \( x = 0 \). In the slope-intercept form (\( y = mx + b \)), the y-intercept is denoted as \( b \).

• This point is significant because it gives an anchor point for drawing the graph of the line.
• It also provides an immediate sense of where the line sits on the graph, irrespective of the slope.

For the function \( f(x) = -0.5x - 0.2 \), the y-intercept is \(-0.2\). Here's what you can deduce:

• When \( x = 0 \), the value of \( y \) is \(-0.2\).
• This tells you the exact point where the line meets the y-axis.
• Graphically, it's the starting point from which you can plot additional points using the slope.

By plotting the y-intercept first and then using the slope to find other points, you can draw the entire line accurately on a graph.