To determine which function represents the height, [tex]\( y \)[/tex], of the railing in inches according to the horizontal distance in inches, [tex]\( x \)[/tex], from the top of the stairs, we need to consider the nature of the problem.
Given that the height of the railing, [tex]\( y \)[/tex], decreases as the horizontal distance, [tex]\( x \)[/tex], increases, we should look for a function with a negative slope. This means that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] should decrease.
Here are the given functions:
1. [tex]\( y = -\frac{3}{4} x + 36 \)[/tex]
2. [tex]\( y = -3 x + 36 \)[/tex]
3. [tex]\( y = \frac{3}{4} x + 36 \)[/tex]
4. [tex]\( y = 3 x + 36 \)[/tex]
Let's analyze each function:
1. [tex]\( y = -\frac{3}{4} x + 36 \)[/tex]:
- The slope here is [tex]\(-\frac{3}{4}\)[/tex].
- Since the slope is negative, [tex]\( y \)[/tex] decreases as [tex]\( x \)[/tex] increases.
2. [tex]\( y = -3 x + 36 \)[/tex]:
- The slope here is [tex]\(-3\)[/tex].
- Since the slope is negative, [tex]\( y \)[/tex] decreases as [tex]\( x \)[/tex] increases.
3. [tex]\( y = \frac{3}{4} x + 36 \)[/tex]:
- The slope here is [tex]\(\frac{3}{4}\)[/tex].
- Since the slope is positive, [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] increases.
4. [tex]\( y = 3 x + 36 \)[/tex]:
- The slope here is [tex]\(3\)[/tex].
- Since the slope is positive, [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] increases.
From our analysis, functions 1 and 2 have negative slopes, meaning they both fit the criterion that [tex]\( y \)[/tex] decreases as [tex]\( x \)[/tex] increases. However, we need to determine which one is the most appropriate.
Given the additional verification process of determining values, our focus is set on the first function:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
This function represents the height, [tex]\( y \)[/tex], of the railing in inches according to the horizontal distance in inches, [tex]\( x \)[/tex], from the top of the stairs.
Thus, the function that can represent the height, [tex]\( y \)[/tex], of the railing in inches according to the horizontal distance in inches, [tex]\( x \)[/tex], from the top of the stairs is:
[tex]\[ y = -\frac{3}{4} x + 36 \][/tex]
