Problem 29 Identify the slope and $y$ -in... [FREE SOLUTION]

Chapter 5: Problem 29

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

Short Answer

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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 by step solution

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: $$ m = -0.5 $$

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: $$ b = -0.2 $$

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 $f(x) = -0.5x - 0.2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Identify the slope

Identify the y-intercept

Graph the function

Key Concepts

Slope-Intercept Form

Slope

Y-Intercept

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