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Chapter 1: Problem 49

Use the y-intercept and slope to sketch the graph of each equation. $$y=-3 x+1$$

Short Answer

Expert verified
Plot y-intercept (0,1), use slope -3 to find point (1,-2), and draw the line.

Step by step solution

01

Identify the Slope and Y-Intercept

The equation of the line is given in the slope-intercept form, which is y = mx + b. Here, 'm' is the slope and 'b' is the y-intercept. In the equation y = -3x + 1, the slope (m) is -3 and the y-intercept (b) is 1.
02

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For the equation y=-3x+1, the y-intercept is 1, so plot the point (0, 1) on the graph.
03

Use the Slope to Find Another Point

The slope of the line is -3, which means for every 1 unit you move to the right along the x-axis, the value of y decreases by 3 units (since the slope is negative). Starting from the y-intercept (0, 1), move 1 unit to the right to (1, 1) and then move 3 units down to plot the point (1, -2).
04

Draw the Line

With the two points (0, 1) and (1, -2) plotted, draw a straight line through these points. Extend the line in both directions to complete the graph.

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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 of a linear equation is one of the most common and easiest ways to represent a line. This form is written as \(y = mx + b\). Here鈥檚 a breakdown of each component:
  • \(y\): This is the dependent variable or the value on the y-axis.
  • \(m\): This represents the slope of the line. The slope indicates how steep the line is, and it is defined as the 'rise' over 'run' or \(\frac{\Delta y}{\Delta x}\).
  • \(x\): This represents the independent variable or the value on the x-axis.
  • \(b\): This is the y-intercept. It鈥檚 where the line crosses the y-axis when \(x = 0\).

In the given equation \(y = -3x + 1\), the slope \(m\) is -3 and the y-intercept \(b\) is 1. By recognizing this form, we can plot the line easily.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. It鈥檚 crucial for plotting a graph because it gives us a starting point.
In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\).

For our equation \(y = -3x + 1\), the y-intercept is 1. This means the line crosses the y-axis at the point (0,1).

To plot the y-intercept:
  • Find the point where the line will intersect the y-axis, which is where \(x = 0\).
  • In our equation, when \(x = 0\), \(y = 1\).
  • So, we plot the point (0,1) on the graph.

This gives us the initial point to draw our line, and in the next step, we will use the slope to find other points.
Graphing Lines
Graphing a line involves plotting points and then drawing a line through those points. To graph lines using the slope and y-intercept method, follow these steps:
  • Identify the slope \(m\) and the y-intercept \(b\).
  • Plot the y-intercept point on the graph.
  • Use the slope to find another point on the line. Recall that the slope \(m\) can be seen as \(\frac{rise}{run}\).
  • Starting from the y-intercept, use the slope. For \(y = -3x + 1\), the slope is -3, meaning for every 1 unit you move right (run), move 3 units down (rise).
  • Plot the second point using slope calculations, in this case, starting at (0,1) and moving to (1,-2).

Finally, draw a straight line through these points and extend it in both directions to complete your graph.
Graphing lines with the slope-intercept method is straightforward and a fundamental skill in algebra.

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