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Chapter 0: Problem 36

Sketching a Line in the Plane In Exercises \(31-38,\) sketch a graph of the equation. $$ y-1=3(x+4) $$

Short Answer

Expert verified
The plot of the equation \(y - 1 = 3(x + 4)\) is a straight line with a slope of 3 and a y-intercept of 13.

Step by step solution

01

Change the equation into slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope and b is the y-intercept. Starting with the equation \(y - 1 = 3(x + 4)\), distribute the 3 on the right side to get \(y - 1 = 3x + 12\). Then, add 1 to both sides to isolate y and get the equation into slope-intercept form for a linear equation, \(y = 3x + 13\).
02

Identify the slope and the y-intercept

In the equation \(y = 3x + 13\), the coefficient of x, 3, is the slope (m) and 13 is the y-intercept (b). This tells you that for every step you move to the right along the x-axis, you'll move 3 steps up along the y-axis. Also, the line intersects the y-axis at point (0, 13).
03

Plot the line

First, plot the y-intercept by making a point at (0, 13) on the graph. Then, use the slope to find another point on the line. Move 1 step to the right and 3 steps up from the y-intercept, and make a second point at (1, 16). Then, draw the line that passes through these two points.

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Key Concepts

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

Slope-Intercept Form
When dealing with linear equations, the slope-intercept form is quite handy. It makes graphing and understanding lines easier. This form is written as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. Slope-intercept form simplifies equations and reveals key information at a glance. By rewriting equations in this structure, you clearly see how the line behaves. In the given exercise, convert \( y - 1 = 3(x + 4) \) to \( y = 3x + 13 \). This form immediately tells you how steep the line is and where it crosses the y-axis. Understanding how to transform an equation into slope-intercept form sets the stage for effective graphing.
Linear Equation
A linear equation, as the name suggests, describes a straight line on a graph. Its general form is \( Ax + By = C \), but you often see it in slope-intercept form for easier graphing. Linear equations maintain a constant rate of change. For example, in the equation \( y = 3x + 13 \), each change in \( x \) results in a proportional change in \( y \). You can recognize these equations by their highest exponent: it should always be one. Linear equations are the backbone of graphing activities, making it essential to identify and work comfortably with them.
Graphing a Line
Graphing a line starts with identifying important components of the equation. Begin by pinpointing the y-intercept and slope from the slope-intercept form. For our exercise, the equation was simplified to \( y = 3x + 13 \).
  • First, plot the y-intercept \((0, 13)\) on the graph.
  • Then use the slope to locate a second point by moving 1 unit right and 3 units up, landing at \((1, 16)\).
Connecting these points with a straight line completes the process. Ensure the line extends across the graph for a comprehensive representation. Carefully plotting is crucial for accurate results and demonstrates the linear equation in visual form.
Slope
Slope is a measure of the steepness of a line, expressed as rise over run. In the equation \( y = mx + b \), the coefficient \( m \) is the slope. For the line \( y = 3x + 13 \), the slope is 3.
  • It tells you that with each unit increase in \( x \), \( y \) increases by 3 units.
  • This predictable change helps in plotting points and drawing lines.
Understanding slope allows you to gauge how steep a line is and how quickly it ascends or descends. It鈥檚 crucial for determining the direction and angle of a line on a graph.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis. In slope-intercept form, \( y = mx + b \), \( b \) is the y-intercept. From our example \( y = 3x + 13 \), the y-intercept is 13.
  • This means when \( x \) is zero, \( y \) is 13.
  • The point is plotted at \((0, 13)\).
This intercept is crucial for initial graphing. It provides a starting point for plotting the line and is foundational for graphing using the slope-intercept approach. Recognizing and using the y-intercept simplifies drawing accurate graphs.

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