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Chapter 4: Problem 314

Graph the line of each equation using its slope and \(y\) -intercept. \(y=-\frac{3}{5} x+2\)

Short Answer

Expert verified
Identify slope and y-intercept, plot them, and draw the line.

Step by step solution

01

Identify the Slope and y-intercept

Given the equation of the line in slope-intercept form, which is \(y = mx + b\), identify the slope \(m\) and the y-intercept \(b\). For the given equation \(y = -\frac{3}{5} x + 2\), the slope \(m\) is \(-\frac{3}{5}\) and the y-intercept \(b\) is 2.
02

Plot the y-intercept

On the Cartesian plane, plot the point where the line crosses the y-axis. This is at \(y = 2\). So, plot the point \((0, 2)\).
03

Use the Slope to Find Another Point

Starting from the y-intercept \((0, 2)\), use the slope \(-\frac{3}{5}\) to find another point. The slope \(-\frac{3}{5}\) means that for every 5 units you move horizontally (to the right), you move 3 units vertically (downward) because the slope is negative. From \((0, 2)\), move 5 units to the right to \((5, 2)\), then move 3 units down to \((5, -1)\). Plot the point \((5, -1)\).
04

Draw the Line

Draw a straight line through the points \((0, 2)\) and \((5, -1)\). This is the graph of the given equation.

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Key Concepts

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

slope-intercept form
Understanding the slope-intercept form is essential for graphing linear equations. The general formula is written as: \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, where the line crosses the y-axis.
The slope-intercept form makes it easy to quickly identify key characteristics of the line:
  • Slope \(m\): This tells you how steep the line is. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.
  • Y-intercept \(b\): This is the value of y where the line crosses the y-axis (when \(x = 0\)).
    For the equation \(y = -\frac{3}{5} x + 2\), the slope is \(-\frac{3}{5}\), and the y-intercept is 2.
plotting points
Plotting points accurately is crucial when graphing linear equations. Here鈥檚 a simple guide to plot points on a Cartesian plane:
Identify the coordinates: Each point is presented as \((x, y)\).
  • Begin with the y-intercept: In our example \(y = -\frac{3}{5} x + 2\), the y-intercept is 2. So, start by plotting the point \((0, 2)\).
  • Use the slope for the next point: The slope \(-\frac{3}{5}\) indicates a rise over run. From the y-intercept, move 5 units to the right (positive x-direction) and 3 units down (negative y-direction) due to the negative slope. This gives us the point \((5, -1)\).
Plotting points accurately helps visualize the equation, making it easier to draw the line and interpret its behavior.
Cartesian plane
The Cartesian plane, or coordinate plane, is a two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis). These lines intersect at the origin \((0,0)\). The plane is divided into four quadrants:
  • Quadrant I: both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: both x and y are negative.
  • Quadrant IV: x is positive, y is negative.
When graphing equations, understanding where to plot points on the Cartesian plane is essential. For example, the point \( (0, 2)\) is in Quadrant I, as both coordinates are positive, while the point \((5, -1)\) lies in Quadrant IV. Learning how to plot points across these quadrants helps in drawing accurate graphs and understanding the spatial relationships between points.
slope
The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y (vertical change) over the change in x (horizontal change). Mathematically, it is expressed as:
\[ m = \frac{\text{change in y}}{\text{change in x}} \]
In our example \( y = -\frac{3}{5} x + 2 \), the slope \(-\frac{3}{5}\) can be interpreted as:
  • -3: move 3 units down (negative vertical direction).
  • 5: move 5 units to the right (positive horizontal direction).
This means for every 5 units you move right, you move 3 units down. Understanding slope is essential for predicting the line鈥檚 behavior and helps in manually drawing the line on a graph from any given point.

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Most popular questions from this chapter

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