Problem 16 - Precalculus Essentials

Question

Graph each equation.Let \(x=-3,-2,-1,0\) \(1,2,\) and 3 $$y=x+2$$

Step-by-Step Solution

Answer

The points that need to be plotted are: (-3,-1), (-2,0), (-1,1), (0,2), (1,3), (2,4), and (3,5). The line passes through all these points and shows the linear relationship represented by the equation \(y=x+2\).

Step 1: Understand the equation

The given equation is in slope-intercept form, \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, the slope is 1 (coefficient of \(x\)), and the y-intercept is 2.

Step 2: Substitute x-values into the equation

Replace \(x\) in the equation with each of the given x-values: -3, -2, -1, 0, 1, 2, and 3. For \(x=-3\), \(y=(-3)+2=-1\). For \(x=-2\), \(y=(-2)+2=0\). For \(x=-1\), \(y=(-1)+2=1\). For \(x=0\), \(y=(0)+2=2\). For \(x=1\), \(y=(1)+2=3\). For \(x=2\), \(y=(2)+2=4\). For \(x=3\), \(y=(3)+2=5\). So the points to plot are: (-3,-1), (-2,0), (-1,1), (0,2), (1,3), (2,4), and (3,5).

Step 3: Plot the points and draw the line

Using a graphing tool, plot each of the points calculated in the previous step. After all points are plotted, draw a line that passes through each of the points. The line should show the linear relationship represented by the equation \(y=x+2\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a straight line. It's written as \( y = mx + b \), where \( m \) represents the slope and \( b \) denotes the y-intercept. This form is incredibly useful because it clearly shows two main characteristics of a linear equation without needing further rearrangement.

• **Slope \( (m) \)**: The slope indicates the steepness of the line and its direction (upwards if the slope is positive, and downwards if negative).
• **Y-intercept \( (b) \)**: This is the point where the line crosses the y-axis. It's simply the value of \( y \) when \( x = 0 \).

In the equation \( y = x + 2 \), the slope \( m \) is 1, showing that for each increase in \( x \) by 1, \( y \) also increases by 1. The y-intercept \( b \) is 2; meaning that the line crosses the y-axis at \( (0, 2) \). This form makes it easy to start graphing because these features are immediately clear.

Plotting Points

Plotting points is a crucial step in graphing any equation, especially linear ones, since it involves placing points on a coordinate grid that satisfy the equation's rules. Here's a simple guide:

- First, take a set of values for \( x \) (like -3, -2, -1, 0, 1, 2, 3 in the exercise) and substitute them into the equation to find corresponding \( y \) values.- Each \( x \) and \( y \) pair indicates a point you can plot. For instance, substituting \( x = -3 \) into \( y = x + 2 \) gives \( y = -1 \), forming the point \((-3, -1)\).- Do this for all chosen values of \( x \) to create several points.- Plot these points on a graph, using a grid where the x-axis is horizontal and the y-axis is vertical.
Once all points are plotted, these will form a visual representation of the equation. Connecting these dots with a line will complete your graph of the linear equation. Each point is like a snapshot showing how \( x \) values affect \( y \) values under the defined rule.

Linear Relationships

Linear relationships are connections between two variables that produce a straight line when graphed. This straightforward relationship is characterized by a constant rate of change or slope. Understanding their properties makes them simpler to analyze:

- **Constant Rate of Change**: The slope of a line, \( m \), is always the same no matter which points you pick along the line. This means there's a uniform increase or decrease in one variable when the other changes.
- **Graph Straight Line**: When you plot points that represent a linear relationship, they should line up perfectly. Drawing a line through these points should result in a straight line.
- **Predictability**: Because linear relationships have constant properties, you can easily predict the value of \( y \) for any given \( x \) using the equation of the line.
In the graph of the equation \( y = x + 2 \), each plotted point lies perfectly along a straight line, demonstrating the predictability and uniformity of a linear relationship. Observing and analyzing these relationships can help solve real-world problems and make calculations more manageable.