Chapter 0: Problem 1

Graph the following equations. $$ y=2 x-1 $$

### Short Answer

Expert verified

Plot y-intercept at (0, -1), use slope 2 to find (1, 1), draw the line through these points.

## Step by step solution

01

## Find the y-intercept

Identify the y-intercept from the equation. The equation is in slope-intercept form, which is given by \(y = mx + b\). The constant term \(b\) represents the y-intercept. In this equation, \(b = -1\). So, the y-intercept is at the point (0, -1).

02

## Determine the slope

In the equation \(y = 2x - 1\), the coefficient of \(x\) is 2. This value represents the slope \(m\). The slope indicates the rise over run and implies that for every increase of 1 unit in the x-direction, y increases by 2 units.

03

## Plot the y-intercept

Start by plotting the y-intercept point (0, -1) on the graph. This is where the line will cross the y-axis.

04

## Use the slope to find another point

From the y-intercept (0, -1), use the slope to determine another point on the line. The slope is 2, meaning from the y-intercept, move 1 unit to the right (positive x-direction) and 2 units up (positive y-direction). This brings you to the point (1, 1).

05

## Plot the second point

Plot the point (1, 1) on the graph. This gives you a second reference point for drawing the line.

06

## Draw the line

Draw a straight line through the two points (0, -1) and (1, 1). Extend the line across the graph to complete the visual representation of the equation \(y = 2x - 1\).

## Key Concepts

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

###### Slope-Intercept Form

The slope-intercept form is an equation of a line written as \(y = mx + b\). This form is very useful because it immediately gives you the slope and the y-intercept of the line.

The letter \(m\) represents the slope, which indicates how steep the line is. The letter \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.Knowing both of these values makes it easy to graph the line.

###### Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is represented by the value of \(b\) in the slope-intercept form equation \(y = mx + b\).

For example, in the equation \(y = 2x - 1\), \(b = -1\). This tells us that the line crosses the y-axis at the point (0, -1). To find the y-intercept, look at the constant term (the number without the variable). Plotting this point on the graph gives you a starting point to draw your line.

###### Slope Calculation

The slope of a line (represented by \(m\)) describes how steep the line is. It is calculated as the 'rise' over the 'run', which is the change in y divided by the change in x.

In the slope-intercept form \(y = mx + b\), the slope is the coefficient of \(x\). For the equation \(y = 2x - 1\), the slope is 2. This means that for every 1 unit you move to the right along the x-axis, the y value increases by 2 units. Positive slopes go upwards, while negative slopes go downwards.

###### Plotting Points

To graph a line, you need at least two points. First, plot the y-intercept on the graph. For the equation \(y = 2x - 1\), this point is (0, -1).

Next, use the slope to find another point. With a slope of 2, from the y-intercept (0, -1), move 1 unit right (positive x-direction) and 2 units up (positive y-direction). This gives you the point (1, 1).

Plot these two points on the graph.

By plotting points, you create a visual reference to draw the line.

###### Linear Graph

A linear graph represents a linear equation and is always a straight line. Once you have plotted two points, you can draw a straight line through them. This line can extend in both directions.

For the equation \(y = 2x - 1\), plot the points (0, -1) and (1, 1), and draw a line through them. This line shows all possible solutions to the equation.

A linear graph is one of the simplest ways to visually understand the relationship between x and y in an equation.

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