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Which graph accurately depicts the correct formula for the points : (1, 5), (2, 7), (3, 9), and (4, 11)

3 answers

To determine the correct formula that represents the points (1, 5), (2, 7), (3, 9), and (4, 11), we can start by looking for a linear relationship.

Let’s analyze the points:

  • When \( x = 1 \), \( y = 5 \)
  • When \( x = 2 \), \( y = 7 \)
  • When \( x = 3 \), \( y = 9 \)
  • When \( x = 4 \), \( y = 11 \)

We can observe that the values of \( y \) increase by 2 for every increase of 1 in \( x \):

\[ \begin{align*} 2 & \text{ (from } 5 \text{ to } 7) \ 2 & \text{ (from } 7 \text{ to } 9) \ 2 & \text{ (from } 9 \text{ to } 11) \ \end{align*} \]

This suggests a linear relationship of the form \( y = mx + b \).

To find the slope \( m \):

\[ m = \frac{\Delta y}{\Delta x} = \frac{2}{1} = 2 \]

Now, we can use one point to find the y-intercept \( b \). Using point (1, 5):

\[ 5 = 2(1) + b \] \[ 5 = 2 + b \] \[ b = 5 - 2 = 3 \]

Thus, the linear equation that describes the points is:

\[ y = 2x + 3 \]

Now, to visualize the graph, we can plot the line \( y = 2x + 3 \) and see if it passes through the given points:

  • For \( x = 1 \): \( y = 2(1) + 3 = 5 \)
  • For \( x = 2 \): \( y = 2(2) + 3 = 7 \)
  • For \( x = 3 \): \( y = 2(3) + 3 = 9 \)
  • For \( x = 4 \): \( y = 2(4) + 3 = 11 \)

Since all points match and are on the line, the graph of the line \( y = 2x + 3 \) accurately represents the data points (1, 5), (2, 7), (3, 9), and (4, 11).

If you can provide the graphs or options to choose from, I can help identify which one accurately represents this equation.

ok but what will the graph look like

To visualize the graph of the equation \( y = 2x + 3 \), we can outline its key features:

  1. Slope: The slope \( m = 2 \) means that for every increase of 1 in \( x \), \( y \) increases by 2. This creates an upward slant from left to right.

  2. Y-intercept: The y-intercept \( b = 3 \) indicates that the line crosses the y-axis at the point (0, 3).

Using this information, we can plot the line. Here's how the graph will look:

  • Coordinate Points:
    • When \( x = 0 \), \( y = 3 \) → Point (0, 3)
    • When \( x = 1 \), \( y = 5 \) → Point (1, 5)
    • When \( x = 2 \), \( y = 7 \) → Point (2, 7)
    • When \( x = 3 \), \( y = 9 \) → Point (3, 9)
    • When \( x = 4 \), \( y = 11 \) → Point (4, 11)
  1. Line: Draw a straight line that passes through these points.

Visualization Steps

  1. Draw Axes: Draw a horizontal axis (x-axis) and a vertical axis (y-axis).
  2. Mark Points: Plot the points (0, 3), (1, 5), (2, 7), (3, 9), and (4, 11) on the graph.
  3. Line: Connect these points with a straight line, extending the line in both directions.

Sample Layout

The graph will look similar to this (text representation):

   |  
11 |                       *
   | 
10 |  
   |  
 9 |                *
   |  
 8 |  
   |  
 7 |         *
   |  
 6 |  
   |  
 5 |    *
   |  
 4 |  
   |  
 3 | * (0, 3)
   |
 2 |  
   |  
 1 |
   |
---+-----------------------
   |  0   1   2   3   4

The points where the asterisks (*) are located represent the points you provided, and the straight line shows the linear relationship described by the equation \( y = 2x + 3 \).

If you plot this on graphing software or with graph paper, you should see it accurately reflect the data points you mentioned.