Algebra/Graphs

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Expert: - 9/21/2008


Question

Diagram
A graph passes through the points (0,a) , (b,c) , and (d,e)

(b,c) is the vertex. What is its equation in terms of a,b,c,d,e and x?

Can a graph be designed to fit these three points?

Diagram is an attachment

Answer
The graph looks like a parabola , so suppose the equation is

y = rx^2 + sx + t

We must figure out what r,s,t must be in terms of a,b,c,d,e to be sure that the given points (0,a) (b,c) (d,e) are on the graph

(0,a) is on the graph so

a = (r)(0^2) + (s)(0)+ t  = t

so t = a


(b,c) is on the graph so

c = (r)(b^2) + (s)(b)+ t = rb^2 + sb + a

rb^2 + sb + a = c

rb^2 + sb = c - a


(d,e) is on the graph so

e = (r)(d^2) + (s)(d)+ t = rd^2 + sd + a

rd^2 + sd + a = e

rd^2 + sd = e - a


Use the two equations to solve for r and s in terms of a,b,c,d and e

rb^2 + sb = c - a
rd^2 + sd = e - a

From the first equation ,

s = (c - a - rb^2)/b

substitute this expression for s into the second equation and solve for r

rd^2 + d(c - a - rb^2)/b = e - a

rd^2 + dc/b - da/b - drb = e - a

r(d^2 - db) = e - a - dc/b + da/b

r(bd^2 - db^2)= be - ba - dc + da

r = (be - ba - dc + da)/ (bd)(d-b)


now substitute this expression for r into

s = (c - a - rb^2)/b


and get


s = (c - a - b^2(be - ba - dc + da)/(bd)(d-b))/b

Simplify to


s = c/b - a/b - b(be - ba - dc + da)/(bd)(d-b)

s = c/b - a/b - (be - ba - dc + da)/(d)(d-b)

s = c/b - a/b - a/d + (dc-be)/(d)(d-b)


The equation is then


y = [(be - ba - dc + da)/ (bd)(d-b)]x^2 +
[c/b - a/b - a/d + (dc-be)/(d)(d-b)]x  + a


The graph will go through the three points.  

Socrates

Expertise

Any questions on High School Algebra, College Algebra, Abstract Algebra. I can help with word problems, solving equations, trigonometry, inequalities, Gaussian Elimination, Linear Algebra, groups, fields, you name it!

Experience

Ph.D. in Mathematics, specialist in Algebra. I have taught High School students and college students at three state universities.

Organizations
Mathematical Association of America. American Mathematical Society.

Publications
Regular contributions to the problems section of the American Mathematical Monthly journal.

Education/Credentials
B.S., M.S., Ph.D.

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