The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown. Which...

1. VISUALIZE the key features of the line

Looking at the graph, identify two critical features:

• Direction of the line: Does it go up or down as you move left to right?
  • This line clearly goes downward from left to right

• Y-intercept location: Where does the line cross the y-axis?
  • The line crosses the y-axis at approximately y = 13.5
  • This is above zero, so it's a positive value

2. INFER what these features mean algebraically

Now connect what you see to the equation form \(\mathrm{y = a + bx}\):

• Downward direction means:
  • The slope is negative
  • In \(\mathrm{y = a + bx}\), we need 'b' to be negative
  • This means we need a minus sign before the x term

• Positive y-intercept (13.5) means:
  • The constant 'a' should be +13.5
  • Not -13.5

3. TRANSLATE these requirements to the answer choices

Looking for: positive constant term (13.5) AND negative coefficient on x (-0.8)

• Choice A: \(\mathrm{y = 13.5 + 0.8x}\)
  • Y-intercept: ✓ positive (13.5)
  • Slope: ✗ positive (the \(\mathrm{+0.8x}\) means slope is +0.8, so line goes up)

• Choice B: \(\mathrm{y = 13.5 - 0.8x}\)
  • Y-intercept: ✓ positive (13.5)
  • Slope: ✓ negative (the \(\mathrm{-0.8x}\) means slope is -0.8, so line goes down)

• Choice C: \(\mathrm{y = -13.5 + 0.8x}\)
  • Y-intercept: ✗ negative (-13.5 would cross below the x-axis)
  • Slope: ✗ positive

• Choice D: \(\mathrm{y = -13.5 - 0.8x}\)
  • Y-intercept: ✗ negative (-13.5)
  • Slope: ✓ negative

Only Choice B matches both requirements!

Answer: B (\(\mathrm{y = 13.5 - 0.8x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing which part of the equation represents which feature of the line

Students often mix up the roles of the constant term and the coefficient:

• They might think the sign in front of x only affects the x-term's value, not realizing it determines whether the entire line goes up or down
• Or they focus on calculating the exact slope from two points but make arithmetic errors with negative numbers

This confusion might lead them to select Choice A (\(\mathrm{y = 13.5 + 0.8x}\)) because they correctly identify the y-intercept as 13.5 but fail to recognize that \(\mathrm{+0.8x}\) gives a positive (upward) slope when the graph clearly shows a downward slope.

Second Most Common Error:

Poor VISUALIZE skill: Misidentifying where the line crosses the y-axis

Some students rush and don't carefully observe where the line intersects the y-axis. They might:

• Look at where data points cluster rather than where the line itself crosses
• Misread the scale and think the y-intercept is negative

This may lead them to select Choice D (\(\mathrm{y = -13.5 - 0.8x}\)) because they correctly identify the negative slope but incorrectly think the y-intercept is negative.

The Bottom Line:

This problem tests your ability to connect visual features of a graph to algebraic notation. The key is systematically checking both the y-intercept (positive or negative?) and the slope (upward or downward?) before selecting your answer. Don't just look at the numbers—pay attention to all the signs!