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Chapter 2: Problem 3

Graph \(y=x+1, y=x+2,\) and \(y=x+3\) on the same rectangular coordinate system.How do the graphs differ?

### Short Answer

Expert verified

The graphs differ in their y-intercepts: 1, 2, and 3 respectively, leading to parallel lines shifted vertically.

## Step by step solution

01

## Understand the Equation's Form

Recognize that all three equations are in the form of a linear function, specifically, they have the form of a straight line given by the equation: y=mx+c where 'm' is the slope and 'c' is the y-intercept. In the given equations, the slope 'm' is 1 for all three lines; thus, they all have the same slope.

02

## Identify the Y-Intercepts

Identify the y-intercepts from the equations: yp =+ : + :1, y =+=2, and y =+=3. So, the y-intercepts are 1, 2, and 3 respectively.

03

## Graph the Equations

Graph each equation on the same rectangular coordinate system: For y = x + 1: The line crosses the y-axis at 1. For y = x + 2: The line crosses the y-axis at 2. For y = x + 3: The line crosses the y-axis at 3. All three lines will be parallel to each other because they all have the same slope.

04

## Analyze the Differences

Since all three lines have the same slope (m=1), they are parallel. The only difference is their y-intercepts, which shift each line vertically: The graph of y = x + 1 is shifted 1 unit up from y = x. The graph of y = x + 2 is shifted 1 unit up from y = x + 1. The graph of y = x + 3 is shifted 1 unit up from y = x + 2.

## Key Concepts

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

###### linear functions

Linear functions are mathematical expressions that result in straight-line graphs. Their general form is given by the equation \[ y = mx + c \], where 'm' is the slope and 'c' is the y-intercept. These functions describe a constant rate of change, which means the relationship between the x and y variables is always proportional. When you graph linear functions, you'll notice that the line never curves. It's always straight, indicating a steady increase or decrease.

###### slope-intercept form

The slope-intercept form of a linear equation is one of the most common and useful forms. It is written as \[ y = mx + c \], where 'm' stands for the slope and 'c' represents the y-intercept.

The slope 'm' tells us how steep the line is, showing how much y changes for a unit change in x. For every increase of 1 in x, y increases by 'm'.

The y-intercept 'c' is the point where the line crosses the y-axis. This form is especially handy for quickly graphing a linear equation and understanding its basic characteristics.

###### parallel lines

Parallel lines are lines that never intersect, no matter how far they are extended. In the context of linear equations, this happens when two or more lines have the same slope.

In our exercise, the lines given by the equations \[ y = x + 1 \], \[ y = x + 2 \], and \[ y = x + 3 \] all have the same slope of 1. This means they will be parallel to each other. The only difference between them lies in their y-intercepts. While they are shifted vertically due to different y-intercepts, their direction and steepness remain the same, indicating parallelism.

###### y-intercept

The y-intercept is where a linear function crosses the y-axis. It is represented by the value 'c' in the slope-intercept form of an equation \[ y = mx + c \].

For the equations in our exercise \[ y = x + 1 \], \[ y = x + 2 \], and \[ y = x + 3 \], the y-intercepts are 1, 2, and 3 respectively. This causes the lines to start at different points on the y-axis while maintaining the same slope. Understanding the y-intercept helps determine where the line will begin vertically, making it easier to graph the equation accurately.

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