Question

How do you find a quadratic function that passes through the points `(1, 4)`, `(-1, 6)` and `(-2, 16)` ?

Answer 1

Function is `y=3x^2-x+2`

Explanation:

Let the quadratic function be `y=ax^2+bx+c`. As it passes through `(1,4)`, `(-1,6)` and `-2,16`, we will have

`a(1)^2+b*1+c=4` i.e. `a+b+c=4` ................................(1)

`a(-1)^2+b*(-1)+c=6` i.e. `a-b+c=6` ................................(2)

`a(-2)^2+b*(-2)+c=6` i.e. `4a-2b+c=16` ................................(3)

Adding (1) and (2) gives us `2a+2c=10`

i.e. `a+c=5` and `b=-1`................................(4)

Multiplying (1) by `2` and adding it to (3), we get

`2a+2b+2c+4a-2b+c=2xx4+16`

or `6a+3c=24` i.e. `2a+c=8` ................................(5)

Subtracting (4) from (5) `a=3` and hence `c=2`

and function is `y=3x^2-x+2`
graph{3x^2-x+2 [-3, 3, -2, 18]}

Answer 2

`f(x) = 3x^2-x+2`

Explanation:

Given:

`(1, 4)`, `(-1, 6), (-2, 16)`

We can immediately write down a formula for a quadratic that goes through these points by constructing terms for each distinct value of `x` we want to match:

`f(x) = color(blue)(4)*((x-(color(blue)(-1)))(x-(color(blue)(-2))))/(((color(blue)(1))-(color(blue)(-1)))((color(blue)(1))-(color(blue)(-2)))) +`

`color(white)(f(x) =) color(blue)(6)*((x-(color(blue)(1)))(x-(color(blue)(-2))))/(((color(blue)(-1))-(color(blue)(1)))((color(blue)(-1))-(color(blue)(-2)))) +`

`color(white)(f(x) =) color(blue)(16)*((x-(color(blue)(1)))(x-(color(blue)(-1))))/(((color(blue)(-2))-(color(blue)(1)))((color(blue)(-2))-(color(blue)(-1))))`

`color(white)(f(x)) = color(blue)(4) * ((x+1)(x+2))/((2)(3)) + color(blue)(6) * ((x-1)(x+2))/((-2)(1)) + color(blue)(16) * ((x-1)(x+1))/((-3)(-1))`

`color(white)(f(x)) = 2/3 (x^2+3x+2) - 3 (x^2+x-2) + 16/3(x^2-1)`

`color(white)(f(x)) = 3x^2-x+2`

This works by adding together the desired multiples of quadratics that take the value `1` at a given `x` coordinate and `0` at the others.

graph{(y-(3x^2-x+2))(8(x-1)^2+(y-4)^2-0.02)(8(x+1)^2+(y-6)^2-0.02)(8(x+2)^2+(y-16)^2-0.02)=0 [-5.35, 4.7, -3, 20]}

In general, given three points `(x_1, y_1)`, `(x_2, y_2)` and `(x_3, y_3)` with `x_1, x_2, x_3` all distinct, a quadratic function through them is given by:

`f(x) = y_1 ((x-x_2)(x-x_3))/((x_1-x_2)(x_1-x_3)) + y_2 ((x-x_1)(x-x_3))/((x_2-x_1)(x_2-x_3)) + y_3 ((x-x_1)(x-x_2))/((x_3-x_1)(x_3-x_2))`