Problem 20 Graph each equation in Exercises... [FREE SOLUTION]

Chapter 1: Problem 20

Graph each equation in Exercises $13-28$. Let $x=-3,-2,-1,0$ $1,2,$ and 3 $$ y=-\frac{1}{2} x+2 $$

Short Answer

Expert verified

The line nicely goes through points at (-3, 3.5), (-2, 3), (-1, 2.5), (0, 2), (1, 1.5), (2, 1), (3, 0.5).

Step by step solution

Calculate y-values

Using the equation $y=-\frac{1}{2} x +2$, calculate the corresponding y-values for the given x-values. Substituting each x-values and calculating as follows: - For $x = -3$, $y = -\frac{1}{2}*(-3) +2 = 1.5 + 2 = 3.5$- For $x = -2$, $y = -\frac{1}{2}*(-2) +2 = 1 + 2 = 3$- For $x = -1$, $y = -\frac{1}{2}*(-1) +2 = 0.5 + 2 = 2.5$- For $x = 0$, $y = -\frac{1}{2}*(0) +2 = 2$- For $x = 1$, $y = -\frac{1}{2}*(1) +2 = -0.5 + 2 = 1.5$- For $x = 2$, $y = -\frac{1}{2}*(2) +2 = -1 + 2 = 1$- For $x = 3$, $y = -\frac{1}{2}*(3) +2 = -1.5 + 2 = 0.5$

Plotting the points on Graph

Next, plot each ordered pair of x-values and their corresponding y-values on the graph. These points are: (-3, 3.5), (-2, 3), (-1, 2.5), (0, 2), (1, 1.5), (2, 1), (3, 0.5). Notice that these points form a straight line, verifying the given equation is indeed, a linear function.

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Key Concepts

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

Linear Functions

Linear functions form a foundational concept in mathematics, defined by equations that result in straight lines when graphed. These functions have a constant rate of change, meaning the slope is consistent throughout the line. A general form of a linear function is given by $y = mx + b$, where:

$m$ is the slope, representing the steepness and direction of the line.
$b$ is the y-intercept, indicating where the line crosses the y-axis.

They are called "linear" because the graph of such an equation is always a line. The linear function helps in understanding relationships where enlarging the x-value by one unit increases or decreases the y-value by a consistent amount. This consistency makes linear functions simple to work with and apply in various contexts.

Coordinate Graphing

Coordinate graphing involves plotting points on a plane defined by an x-axis and y-axis, often referred to as the Cartesian coordinate system. Each point is represented by an ordered pair $(x, y)$. Here's how it works:

The x-value tells how far to move horizontally from the origin (the point $0, 0$).
The y-value indicates how much to move vertically.

By connecting these plotted points, you can visualize the entire function. Whether they form lines, curves, or other shapes, understanding how to graph coordinates is a critical skill for interpreting mathematical relationships. Coordinate graphing is a visual way to assess and analyze how changes in variables affect outcomes.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write the equation of a line, defined as $y = mx + b$. In this format, you can directly identify both the slope ($m$) and y-intercept ($b$). Here's what these components mean:

Slope ($m$): This tells us how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
Y-intercept ($b$): This is the point where the line crosses the y-axis. It shows the value of $y$ when $x = 0$.

This form is extremely useful in quickly understanding and graphing the behavior of linear equations. By rearranging a linear equation to this form, it becomes easy to plot the line and interpret its characteristics without extensive calculations.

Plotting Points

Plotting points on a graph is an essential skill for visualizing and analyzing equations. Here's a simple guide to getting it right:

First, calculate the coordinates for each point based on given values, as seen in our example equation $y = -\frac{1}{2}x + 2$.
Use an x-value to find its corresponding y-value and create ordered pairs $(x, y)$.
On the graph, mark these points by locating them on the relevant x and y positions.

Once all points are placed, draw a line through them. If the equation is linear, the points will align perfectly to form a straight line. This process not only helps with graphing but also aids in understanding the way different equations map onto graphs, making it easier to comprehend their dynamics.

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91影视

Graph each equation in Exercises \(13-28\). Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$ y=-\frac{1}{2} x+2 $$

Short Answer

Step by step solution

Calculate y-values

Plotting the points on Graph

Key Concepts

Linear Functions

Coordinate Graphing

Slope-Intercept Form

Plotting Points

One App. One Place for Learning.

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