To match the regression equation $\hat{y} = 0.00114x + 2.53$ with the appropriate graph, you should look for a graph that exhibits the following characteristics:

1. Slope:

• The slope of the line is $0.00114$, which is a small positive number. This means the line should have a slight upward tilt, but the increase in $y$ is very gradual as $x$ increases.

2. Y-intercept:

• The y-intercept is $2.53$. This indicates that the line crosses the y-axis at $y = 2.53$ when $x = 0$. The graph should start at this point on the y-axis.

3. Scale:

• Due to the small slope, changes in $y$ will be minimal even for larger values of $x$, so the line will look almost flat if the scale of the $x$-axis spans a large range.

Graph Features:

• A straight line starting at $y = 2.53$ on the y-axis.
• A very gradual upward trend as $x$ increases.

If you have several graphs to choose from, the one with a line that appears almost flat and starts around $2.53$ on the y-axis will be the correct one. The larger the range of $x$, the flatter the line will appear due to the small slope.

Would you like help identifying the correct graph from specific options, or should I further explain how to interpret the graph?

5 Related Questions:

1. How does the slope value affect the appearance of the graph?
2. What happens to the graph if we change the y-intercept value?
3. How do you interpret a very small slope in a regression equation?
4. What real-life situations might have regression lines with small slopes?
5. How would the graph change if the slope were negative instead?

Tip: Always pay close attention to the scale of the axes when interpreting or drawing regression lines—it can greatly affect the perception of the slope's steepness.