Question

which equation could represent the relationship shown in the scatter plot?\
\\(\circ\\) \\(y = 3x\\)\
\\(\circ\\) \\(y = \frac{1}{2}x + 7\\)\
\\(\circ\\) \\(y = \frac{5}{2}x - 1\\)\
\\(\circ\\) \\(y = -x + 8\\)

Explanation:

Step1: Analyze the trend of the scatter plot

The scatter plot shows a positive linear trend, so the slope of the line should be positive (eliminate \( y = -x + 8 \) since its slope is -1).

Step2: Analyze the slope and y - intercept of each remaining equation

• For \( y = 3x \): The slope is 3, which is relatively steep. Looking at the scatter plot, the rate of increase of y with respect to x is not that steep. For example, when \( x = 1 \), \( y=3(1) = 3 \), but the first point has \( x = 1 \) and \( y\approx9 \), so this equation does not fit.
• For \( y=\frac{1}{2}x + 7 \): The slope is \( \frac{1}{2} \) (0.5) and the y - intercept is 7. Let's check some approximate points. When \( x = 1 \), \( y=\frac{1}{2}(1)+7=7.5 \), close to the first point (x = 1, y≈9? Wait, maybe my initial check is wrong. Wait, let's check another point. When \( x = 10 \), \( y=\frac{1}{2}(10)+7=5 + 7=12 \). Looking at the scatter plot, when x is around 10, y is around 11 - 13, which is reasonable.
• For \( y=\frac{5}{2}x-1 \): The slope is \( \frac{5}{2}=2.5 \), which is steeper. When \( x = 4 \), \( y=\frac{5}{2}(4)-1=10 - 1 = 9 \). But in the scatter plot, when x = 4, the y - value is around 12, so this equation does not fit.

Answer:

\( y=\frac{1}{2}x + 7 \)