Which parabola?
Can you work out the equations of all these mixed up parabolas?
Age
16 to 18
Challenge level
Problem
Image
Given that two parabolas have the equations
$$y = x^2 -12x +27$$
and
$$y = -x^2+12x-36$$
can you find the equations of the other parabolas?
This is an Underground Mathematics resource.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Student Solutions
Isobel from Wycombe High School has first identified the given equations as L and K, and then obtained I and J by reflecting these two in the x-axis, and finally obtained the remaining equations by translating I,J,K,L to the left by 6 for E,F,G,H and by 12 for B,A,C,D. This gives the following expressions:
A: $y=-(-(x+12)^2+12(x+12)-36)$
B: $y=-((x+12)^2-12(x+12)+27)$
C: $y=-(x+12)^2+12(x+12)-36$
D: $y=(x+12)^2-12(x+12)+27$
E: $y=-((x+6)^2-12(x+6)+27)$
F: $y=-(-(x+6)^2+12(x+6)-36)$
G: $y=-(x+6)^2+12(x+6)-36$
H: $y=(x+6)^2-12(x+6)+27$
I: $y=-(x^2-12x+27)$
J: $y=-(-x^2+12x-36)$
K: $y=-x^2+12x-36$
L: $y=x^2-12x+27$
When expanded out this gives the simplified equations for the parabolas as:
A: $y=x^2+12x+36$
B: $y=-x^2-12x-27$
C: $y=-x^2-12x-36$
D: $y=x^2+12x+27$
E: $y=-x^2+9$
F: $y=x^2$
G: $y=-x^2$
H: $y=x^2-9$
I: $y=-x^2+12x-27$
J: $y=x^2-12x+36$
K: $y=-x^2+12x-36$
L: $y=x^2-12x+27$
(Another way of getting B,A,C,D is by reflecting I,J,K,L in the y-axis instead of translating to the left by 12)