# How do you write a function rule for x = 2, 4, 6 and y = 1, 0, -1?

Mar 20, 2018

$y = f \left(x\right) = - \frac{1}{2} x + 2$

#### Explanation:

Set the ${i}^{\text{th}}$ point as: ${P}_{i} \to \left({x}_{i} , {y}_{i}\right)$

Change in $x$ sequence is $2$
Change in $y$ sequence is $- 1$

Gradient (slope) $\to m = \left(\text{change in "y)/("change in } x\right) = - \frac{1}{2}$

Assuming there is a direct link between $x \mathmr{and} y$ numbers of sequence we have:

${P}_{1} \to \left({x}_{1} , {y}_{1}\right) = \left(2 , 1\right)$
${P}_{2} \to \left({x}_{2} , {y}_{2}\right) = \left(4 , 0\right)$
${P}_{3} \to \left({x}_{3} , {y}_{3}\right) \to \left(6 , - 1\right)$

Relating $m$ to, say, point 2 we have:

$m = - \frac{1}{2} = \frac{y - {y}_{2}}{x - {x}_{2}} = \frac{y - 0}{x - 4}$

$- \frac{1}{2} = \frac{y - 0}{x - 4}$

Multiply both sides by $\left(+ 2\right)$

$- 1 = \frac{2 \left(y - 0\right)}{x - 4}$

Multiply both sides by $\left(x - 4\right)$

$- \left(x - 4\right) = 2 y$

$- x + 4 = 2 y$

$y = - \frac{1}{2} x + 2$
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The question is specific in that it states 'function rule for $x$'

So we write it as: $y = f \left(x\right) = - \frac{1}{2} x + 2$