Calculus Question!!?

Question

F(x) = { b-x, if x ≤ 1
{a(x-2)^2, if x > 1.
a. Find an equation relating a & b if f is to be continuous at x=1.
b. Find b if a = -1. Show by graphing that f is continuous at x=1 for these values of a & b.
c. Pick another value of a & find b. Show that f is continuous for these values of a & b.

Plz show me how 2 do this problem...I have no clue!...& please tell me how to graph it on the graphing calculator as well.

Answer 1

a.) To find an equation relating a and b to make the function continuous, the easiest way to do that is to use A and B to make both sides of the function equal.

b-x = a(x-2)^2
Let a = 1 to be simple.
b-x = x^2 - 4x + 4 = f(x)
b = x^2 -3x + 4

making f(x) = x^2 - 4x + 4

b.) If a = -a, and the function is continuous, then:
b-x = a(x^2 - 4x + 4)
b-x = -(x^2 - 4x + 4)
b-x= -x^2 +4x - 4 = f(x)
b = -x^2 +5x - 4

c.) Pick another value of a. We'll pick a = x.
b-x = a(x-2)^2
b-x = a(x^2 - 4x + 4)
b-x = x(x^2 - 4x + 4)
b-x = x^3 - 4x^2 + 4x = f(x)
b = x^3 - 4x^2 + 5x

To graph it, graph the f(x) functions that we solved for right before solving for b.

Answer 2

a)

left hand limit at x = 1 is ( b -1)

right hand limit at x = 1 is a

for continuity, left hand limit = right hand limit

so ( b -1) = a or b = a+1

which is the required relation

(b)

if a = -1, b = a+1 = 0.


for graphing, graph y = -x in the interval (-infinity,1]

and graph y = -(x-2)^2 in the interval (1,infinity)

(c)

use any value for a and find b using b = a+1

Answer 3

Good luck

Answer 4

oh my god