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

2 Answers
Nov 6, 2016

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]}

Nov 6, 2016

#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))#