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Chapter 4: Problem 71

For the following exercises, sketch a line with the given features. An \(x\) -intercept (-2,0) and \(y\) -intercept of (0,4)

Short Answer

Expert verified
Draw a line through (-2,0) and (0,4).

Step by step solution

01

Understand the Problem

We need to draw a line that passes through the given intercepts: the x-intercept of (-2,0) and the y-intercept of (0,4). The x-intercept is where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.
02

Write the Equation of the Line

To find the equation of the line, use the slope-intercept form, which is \( y = mx + c \). First, find the slope \( m \) using the two points (x-intercept and y-intercept): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2 \]. Now, substitute one of the points, for example, (0,4), and the slope into the equation: \( 4 = 2 \times 0 + c \) to find c. This gives \( c = 4 \). Therefore, the equation of the line is \( y = 2x + 4 \).
03

Plot the Points on the Graph

Plot the two intercepts on a graph. The first point at \((-2, 0)\) should be on the x-axis, and the second point at \((0, 4)\) should be on the y-axis.
04

Draw the Line

Using a ruler, draw a straight line connecting the two points plotted in Step 3. Extend the line in both directions across the graph. This line represents the equation \( y = 2x + 4 \) with the specified intercepts.

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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 is a way of writing the equation of a line in the format: \( y = mx + c \). Here, \( m \) is the slope of the line, and it measures how steep the line is. It tells you how much \( y \) changes for a change in \( x \). Meanwhile, \( c \) is the \( y \)-intercept, showing where the line crosses the \( y \)-axis. This form is very convenient when you have the slope and the \( y \)-intercept, as it allows quick graphing of the line.
To use this form effectively:
  • Identify the slope \( m \) by calculating the rise over the run between two points on the line.
  • Use any given \( y \)-intercept \( c \) directly to form the equation.
  • Substitute the slope and intercept into the equation structure \( y = mx + c \).
This form helps in gaining insights about the visual representation of the line on a graph and plays a pivotal role in understanding linear relationships.
X-Intercept
The \( x \)-intercept of a line is the point where the line crosses the \( x \)-axis. At this point, the value of \( y \) is zero. Knowing the \( x \)-intercept is quite useful because it gives us a concrete point where the line meets the horizontal axis. This point can be represented as \((-2, 0)\) as in the given exercise. This particular point not only aids in sketching the graph but also is vital during the calculation of the slope.
The concept of the \( x \)-intercept is crucial in:
  • Determining where on the \( x \)-axis the line will meet.
  • Providing one of the two required points to determine the slope.
When given the \( x \)-intercept and the \( y \)-intercept, you can easily progress to sketching the entire line on the graph.
Y-Intercept
The \( y \)-intercept refers to the point where a line crosses the \( y \)-axis. Here, the \( x \) value is always zero. For the exercise at hand, this point is \((0, 4)\). This point is extremely useful because it features directly in the slope-intercept form of the equation, simplifying the equation's formulation.
This intercept serves several purposes:
  • It provides a starting point for graphing a line since it is straightforward to locate on the \( y \)-axis.
  • The \( y \) value of this intercept equals the constant \( c \) in the slope-intercept equation \( y = mx + c \).
Thus, this intercept not only aids in understanding the graphical representation of the line but also simplifies the equation creation process regardless of the slope.
Graphing Linear Equations
Graphing linear equations involves drawing a line on a coordinate grid following the equation format of the line. Once you have your equation in the form of \( y = mx + c \), your next step is to graph it, which requires two main steps:
  • Plot the intercepts mentioned in the equation or given to you.
  • Draw a straight line through the plotted points and extend it in both directions.
For the exercise scenario, after identifying the intercepts, the two points would be plotted: (-2,0) on the x-axis and (0,4) on the y-axis. Connecting these two points with a ruler gives you the visual representation of the line described by the equation. Graphing this way helps in visual understanding and verification of calculations, ensuring that your equation aligns with the observable representation of data.

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Most popular questions from this chapter

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