Question

A linear function, () passes through the points (2, 8) and (5, 17). The function was replaced with ( + ) resulting in the function (). The function () passes through the points (2, 14) and (5, 23). W hat is the value of ?

Answer 1

We can find the original function using the formula

$\dfrac{x-x_2}{x_1-x_2}=\dfrac{y-y_2}{y_1-y_2}$

Plugging your values, we have

$\dfrac{x-5}{2-5}=\dfrac{y-17}{8-17} \iff \dfrac{x-5}{-3}=\dfrac{y-17}{-9}\iff \dfrac{x-5}{3}=\dfrac{y-17}{9}$

Multiply both sides by 9 to get

$y-17=3x-15 \iff y=3x+2$

So, the transformed function is

$y=3(x+k)+2=3x+(2+3k)$

Impose the passing through (2, 14) to get

$14=3\cdot 2+(2+3k) \iff 14=6+2+3k \iff 3k=6 \iff k=2$

So, the new function is

$y=3x+8$