How do you find an equation in standard form of the parabola passing through the points (1,1), (-1, -3), (-3, 1)?

1 Answer
Nov 6, 2016

Please see the explanation for steps leading to the answer:
y=x2+2x2

Explanation:

There are two standard forms for a parabola:

Type 1 opens up or down:

y=ax2+bx+c

Type 2 opens left or right:

x=ay2+by+c

Plot the 3 points:

Here is a plot of the 3 points

The points look like they fit the first type.

Write 3 equations by substituting the 3 points into the type 1 form:

1=a(1)2+b(1)+c [1]
3=a(1)2+b(1)+c [2]
1=a(3)2+b(3)+c [3]

Move the coefficients in front of the variables:

1=a+b+c [1]
3=ab+c [2]
1=9a3b+c [3]

Subtract equation [1] from equation [2]

1=a+b+c [1]
4=2b [2]
1=9a3b+c [3]

1=a+b+c [1]
2=b [2]
1=9a3b+c [3]

Substitute 2 for b into equations [1] and [3]:

1=a+2+c [1]
1=9a3(2)+c [3]

1=a+c [1]
7=9a+c [3]

Subtract equation [1] from equation [3]:

1=a+c [1]
8=8a [3]

1=a+c [1]
1=a [3]

Substitute 1 for a in equation [1]:

c=2

Confirm that the equation: y=x2+2x2 fits all 3 points:

1=(1)2+2(1)2
3=(1)2+2(2)2
1=(3)2+2(3)2

1=1
3=3
1=1

This checks.

The equation is y=x2+2x2