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Chapter 3: Problem 82

Use slope-intercept graphing to graph the equation. $$ y=\frac{2}{5} x-4 $$

Short Answer

Expert verified
Plot (0, -4) and (5, -2) then draw a line through them.

Step by step solution

01

- Identify the slope and y-intercept

The given equation is in the slope-intercept form, which is typically written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. In this equation, \(y = \frac{2}{5} x - 4\), the slope \(m\) is \(\frac{2}{5}\) and the y-intercept \(b\) is -4.
02

- Plot the y-intercept

Locate the y-intercept (0, -4) on the graph. This is the point where the line will cross the y-axis. Place a point at (0, -4).
03

- Use the slope to plot another point

From the y-intercept (0, -4), use the slope \(\frac{2}{5}\). This means you rise 2 units up and run 5 units to the right. So, starting at (0, -4), move up 2 units to reach -2 on the y-axis, then move 5 units to the right to reach (5, -2). Plot this point at (5, -2).
04

- Draw the line

Draw a straight line through the two points, (0, -4) and (5, -2). This line represents the graph of the equation \(y = \frac{2}{5} x - 4\).

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Key Concepts

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

Graphing Linear Equations
Understanding how to graph linear equations is essential in algebra. Linear equations in two variables typically produce straight lines. To graph these equations, it's helpful to use the slope-intercept form. This form makes it easy to identify both the slope and the y-intercept of the equation.
For instance, in the equation \(y = \frac{2}{5} x - 4\), we can immediately see the slope and y-intercept. By following a set of systematic steps, you can plot the equation onto a graph:
  • Identify the y-intercept (the point where the line crosses the y-axis).
  • Use the slope to find another point on the line.
  • Draw a straight line that passes through these points.
This simple process helps make graphing linear equations straightforward and manageable.
Y-Intercept
The y-intercept of a linear equation is the point at which the line crosses the y-axis. It's a crucial part of understanding how the graph looks because it anchors the line in place.
The general formula of a line in slope-intercept form is \(y = mx + b\). Here, the y-intercept is represented by the constant \(b\).
For example, in the equation \(y = \frac{2}{5} x - 4\), the y-intercept is -4. This means that when \(x\) is 0, \(y\) will be -4. To plot this point:
  • Find the point on the y-axis where the y-value is -4.
  • Place a point on the graph at (0, -4).
This point is your starting place for graphing the rest of the equation.
Slope
The slope of a line describes its steepness and direction. It is represented by the coefficient \(m\) in the slope-intercept form equation \(y = mx + b\). The slope is a ratio that compares the vertical change (rise) to the horizontal change (run) between any two points on the line.
In our example, the equation \(y = \frac{2}{5} x - 4\) has a slope of \(\frac{2}{5}\). This means:
  • For every 5 units you move horizontally to the right, the line moves 2 units up vertically.
To use the slope to graph another point:
  • Start from the y-intercept (0, -4).
  • Move 2 units up (rise) to reach -2 on the y-axis.
  • Move 5 units to the right (run) to find the point (5, -2).
By plotting these points, you create a clear path to draw the line representing the equation.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \(\left(-6, \frac{1}{2}\right) ;\left(10, \frac{3}{4}\right)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-7) ;(3,0)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-2,-4) ;(-5,-2)\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-2,-5) ;(1,3)\)

A high-speed Shinkansen train in Japan travels at a speed of \(\frac{270 \mathrm{~km}}{1 \mathrm{hr}}\) for \(18 \mathrm{~min}\). Find the distance it travels.
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