Question

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

Answer 1

ok but what will the graph look like

Answer 2

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.

Answer 3

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)

3. **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.