Question

the graph below shows a relationship between x and y.
which function is the best fit for the data?
$y = -\frac{3}{4}x + 6$
$y = -x + 8$
$y = -\frac{1}{2}x + 7$
$y = -2x + 9$

Explanation:

Step1: Analyze the slope and y-intercept trend

The data points show a negative linear trend. Let's check the approximate slope and y-intercept. When \( x = 0 \) (extrapolating), the y-intercept is around 7. Let's test the slope. For example, when \( x = 1 \), \( y \approx 7.5 \); when \( x = 7 \), \( y \approx 0.5 \). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0.5 - 7.5}{7 - 1}=\frac{-7}{6}\approx - 1.17 \), but let's check the options.

Step2: Test each function

• For \( y = -\frac{3}{4}x + 6 \): At \( x = 1 \), \( y = -\frac{3}{4}(1)+6 = 5.25 \), but actual \( y\approx7.5 \), not close.
• For \( y=-x + 8 \): At \( x = 1 \), \( y = 7 \); at \( x = 7 \), \( y = 1 \). The points seem to align with this.
• For \( y = -\frac{1}{2}x + 7 \): At \( x = 1 \), \( y = 6.5 \), actual \( y\approx7.5 \), less close.
• For \( y=-2x + 9 \): At \( x = 1 \), \( y = 7 \); at \( x = 2 \), \( y = 5 \), but actual \( x = 2 \), \( y\approx5.5 \), but the overall trend of \( y=-x + 8 \) fits better as the slope of -1 and y-intercept 8 matches the general downward trend and the spread of points.

Answer:

\( y = -x + 8 \)