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Chapter 7: Problem 18

Graph each equation using the slope and \(y\) -intercept. $$y=\frac{3}{4} x+2$$

Short Answer

Expert verified
Plot the y-intercept at (0,2), use the slope to find another point, and draw the line through these points.

Step by step solution

01

Identify the Slope and Y-Intercept

The equation given is in the slope-intercept form, which is \(y = mx + c\). Here, the slope \(m\) is \(\frac{3}{4}\) and the y-intercept \(c\) is 2.
02

Plot the Y-Intercept

Begin by plotting the y-intercept on the graph. Since the y-intercept is 2, place a point on the y-axis at \( (0, 2) \).
03

Use the Slope to Find a Second Point

The slope \(\frac{3}{4}\) means for every 4 units you move right on the x-axis, move 3 units up on the y-axis. From the y-intercept \((0, 2)\), move 4 units right to \((4, 2)\) and then 3 units up to \((4, 5)\). Plot this second point \((4, 5)\).
04

Draw the Line

Using a ruler, draw a straight line that passes through both points \((0, 2)\) and \((4, 5)\). Extend the line in both directions, adding arrows to indicate that the line continues indefinitely.

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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 a way to write the equation of a line so you can easily see the slope and y-intercept. It is expressed as \(y = mx + c\) where:
  • \(m\) represents the slope of the line.
  • \(c\) is the y-intercept where the line crosses the y-axis.
This format is incredibly useful because with just a glance, you can identify two crucial pieces of information that help you graph the line on a coordinate plane. By understanding the slope-intercept form, you can easily convert practical scenarios into a mathematical equation, making it a powerful tool for graphing and understanding linear relationships.
Slope
The slope of a line reveals how steep the line is and the direction it moves. In the slope-intercept form \(y = mx + c\), the slope is denoted by \(m\). It tells us two things:
  • If the slope is positive, as in \(\frac{3}{4}\), the line rises from left to right.
  • If the slope is negative, the line falls from left to right.
In our example, the slope \(\frac{3}{4}\) means for every 4 units you move horizontally (x-direction), you move 3 units vertically (y-direction). Slopes help differentiate between lines that are steep versus those that are flat. Understanding slope is essential for determining the angle or direction of a line in a graph.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the format \(y = mx + c\), the y-intercept is represented by \(c\). For the equation \(y = \frac{3}{4}x + 2\), the y-intercept is 2. This means that the line meets the y-axis at the point \((0,2)\).
Plotting the y-intercept is the first step in graphing. It's like a starting point on the graph from which you use the slope to find additional points. By accurately identifying and plotting the y-intercept, you ensure that your line is positioned correctly on the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves to represent mathematical equations visually. It consists of two axes:
  • The x-axis runs horizontally.
  • The y-axis runs vertically.
The point where the x-axis and y-axis intersect is the origin, labeled as \((0, 0)\). To graph a linear equation like \(y = \frac{3}{4} x + 2\), start by plotting the y-intercept on the y-axis, then use the slope to plot more points. Once you have at least two points, you can draw a line through them, extending it across the plane. Understanding how to navigate the coordinate plane is essential for graphing equations and analyzing their relationships visually.

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Most popular questions from this chapter

Draw the graph of a relation that is not a function. Explain why it is not a function.

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