Answered

Which steps should be used to graph the equation below?



$y-4=\frac{1}{3}(x+2)$


A
  1. Plot the point $(2, 4)$.
  2. From that point, count left 3 units and down 1 unit and plot a second point.
  3. Draw a line through the two points.


B
  1. Plot the point $(2, 4)$.
  2. From that point, count left 1 unit and down 3 units and plot a second point.
  3. Draw a line through the two points.


C
  1. Plot the point $(-2, 4)$.
  2. From that point, count left 3 units and down 1 unit and plot a second point.
  3. Draw a line through the two points.

Verified Answer :

  • Identify the point \((-2, 4)\) and slope \(\frac{1}{3}\) from the given equation \(y - 4 = \frac{1}{3}(x + 2)\).
  • Recognize that the slope \(\frac{1}{3}\) can be interpreted as a leftward move of 3 units and a downward move of 1 unit (\(\frac{-1}{-3} = \frac{1}{3}\)).
  • Select the third option, which correctly uses the point \((-2, 4)\) and the movement corresponding to the slope \(\frac{1}{3}\).
  • The correct steps are: 1. Plot the point \((-2, 4)\). 2. From that point, count left 3 units and down 1 unit. 3. Draw the line.

Explanation

  1. Identify Key Components of the Equation

To graph the line given by the equation \(y - 4 = \frac{1}{3}(x + 2)\), we first identify the point and the slope from the point-slope form. The point-slope form of a linear equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Comparing our equation \(y - 4 = \frac{1}{3}(x + 2)\) to the standard form, we identify:

  • \(y_1 = 4\)
  • \(x_1 = -2\)
  • \(m = \frac{1}{3}\)

Thus, a known point on the line is \((-2, 4)\), and the slope is \(\frac{1}{3}\).

  1. Evaluate the Options and Determine the Correct Steps

The slope \(m = \frac{1}{3}\) indicates the rise over the run. This means that for every 3 units we move to the right, we move 1 unit up. Alternatively, the slope can be expressed as \(\frac{-1}{-3}\), which means that for every 3 units we move to the left, we move 1 unit down. Let's evaluate the given options:

  • Option 1 suggests starting at \((2, 4)\). This is incorrect, as the point on our line is \((-2, 4)\).
  • Option 2 suggests starting at \((2, 4)\). This is also incorrect.
  • Option 3 suggests starting at \((-2, 4)\) and moving left 3 units and down 1 unit. Moving left 3 units (a run of -3) and down 1 unit (a rise of -1) results in a slope of \(\frac{-1}{-3} = \frac{1}{3}\), which matches our equation. Therefore, this option correctly identifies the point and the slope.
  1. Conclusion

Based on our analysis, the correct set of steps is:

  1. Plot the point \((-2, 4)\).
  2. From that point, count left 3 units and down 1 unit and plot a second point.
  3. Draw a line through the two points.

This sequence accurately reflects the point \((-2, 4)\) and the slope of \(\frac{1}{3}\) inherent in the equation \(y - 4 = \frac{1}{3}(x + 2)\). Thus, the third set of steps is correct.

Examples

Graphing lines is a foundational skill in algebra that allows us to visualize relationships between variables. For example, if you are planning a road trip and want to represent your distance from home over time as a linear equation, plotting points and using the slope helps you predict when you will reach your destination. By understanding the point-slope form, you can easily map out your journey on a coordinate plane, showing that even if you start at a specific location, you can calculate your path forward or backward based on your speed, or slope. This skill is vital for fields ranging from physics and engineering to economics and data analysis, where visualizing trends is key to making informed decisions.

Related Questions