Question

1. write a function rule to represent each relationship shown by the ordered pairs. \\{(5, -4), (6, -9), (7, -14), (8, -19)\\} \\{(2, -4), (3, 0), (4, 4), (5, 8)\\}

Explanation:

For the set \(\{(5, -4), (6, -9), (7, -14), (8, -19)\}\)

Step1: Find the slope (rate of change)

Let \(x\) be the first element and \(y\) be the second element of the ordered pairs.
Take two points, say \((5, -4)\) and \((6, -9)\).
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-9-(-4)}{6 - 5}=\frac{-9 + 4}{1}=\frac{-5}{1}=- 5\)

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((5,-4)\) and \(m=-5\)
\(y-(-4)=-5(x - 5)\)
\(y + 4=-5x+25\)
\(y=-5x + 21\)
We can check with other points. For \(x = 7\), \(y=-5\times7 + 21=-35 + 21=-14\) (which matches the ordered pair \((7,-14)\)). For \(x = 8\), \(y=-5\times8+21=-40 + 21=-19\) (which matches the ordered pair \((8,-19)\))

For the set \(\{(2,-4),(3,0),(4,4),(5,8)\}\)

Step1: Find the slope (rate of change)

Take two points, say \((2,-4)\) and \((3,0)\)
The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-4)}{3 - 2}=\frac{0 + 4}{1}=4\)

Step2: Use point - slope form \(y - y_1=m(x - x_1)\)

Using the point \((2,-4)\) and \(m = 4\)
\(y-(-4)=4(x - 2)\)
\(y + 4=4x-8\)
\(y=4x-12\)
We can check with other points. For \(x = 4\), \(y=4\times4-12=16 - 12 = 4\) (which matches the ordered pair \((4,4)\)). For \(x=5\), \(y=4\times5-12=20 - 12 = 8\) (which matches the ordered pair \((5,8)\))

Answer:

For \(\{(5, - 4),(6,-9),(7,-14),(8,-19)\}\), the function rule is \(y=-5x + 21\)
For \(\{(2,-4),(3,0),(4,4),(5,8)\}\), the function rule is \(y = 4x-12\)