Question

In your lab, a substance's temperature has been observed to follow the function T(x) = (x - 4)3 + 6. The turning point of the graph is where the substance changes from a liquid to a gas. Explain to your fellow scientists how to find the turning point of this function, using complete sentences.

Answer 1

@phi

Answer 2

Do you know about the turning point formula?

Answer 3

no but since i read your attachment i do now @phi

Answer 4

so A= 4 B=3 C=6 right

Answer 5

I think the a,b, and c are from
a x^3 + b x^2 +c x + d

unfortunately, you have (x - 4)^3 + 6
which is (x-4) (x-4) (x-4) + 6
we have to multiply that out to get it in the form so we can find a, b and c

Answer 6

so to multiple you have distribute

Answer 7

yes. but in steps
first do (x-4)(x-4)

Answer 8

\[x ^{2}-8x +16\]

Answer 9

now do
(x^2 -8x+16)(x-4)

Answer 10

\[x ^{3}-12x ^{2}+16x-64\]

Answer 11

you missed a +32x

Answer 12

oh hold im redoing it

Answer 13

\[x ^{3}-12x ^{2}+48x-64\]

Answer 14

that means you function is
\[ (x - 4)^ 3 + 6 = x ^{3}-12x ^{2}+48x-64 + 6 \\
=x ^{3}-12x ^{2}+48x - 58
\]

Answer 15

ok so now does that means that A=12 B=48 C=-58

Answer 16

almost, you start with x^3 (read that as 1*x^3... so A=1, and so on)

Answer 17

oh i didnt see the x^3

Answer 18

so A=1 B=12 C=48

Answer 19

even closer, but include the signs

Answer 20

yea my bad B=-12

Answer 21

now we use the formula

Answer 22

ok i had written the formula down on a piece of paper

Answer 23

\[-12\sqrt{0} \over 3\]

Answer 24

the formula says -B in front. B is -12, so you have - -12 or +12

Answer 25

and you are adding the sqr(0)
so it is
\[ \frac{12±\sqrt{0}}{3} \]
sqrt(0) is 0 so that is
\[ \frac{12±0}{3} \\\frac{12}{3}

\]

Answer 26

the answer is x=12/3 = 4
that is the turning point of this curve
Here is a graph

Answer 27

ok thank you so much for your help