Question

4. Ray needs help creating the second part of the coaster. Create a unique parabola in the pattern f(x) = ax2 + bx + c. Describe the direction of the parabola and determine the y-intercept and zeros.

Answer 1

My parabola is f(x)= 3x^2+6x+15

Answer 2

Here is the graph https://www.desmos.com/calculator/m9uxxepivl

Answer 3

So would the answer be the parabola is going up because it is positive. And the y-int is 15

Answer 4

@phi

Answer 5

@Sachintha

Answer 6

@AloneS

Answer 7

I dont know about the zeros though

Answer 8

ok on it being U shaped, with a y-intercept of 15
The zeros are where it crosses the x-axis. As you can see, it is well above the x-axis.
You might think it has no zeros, but if we use complex numbers, we can find "zeros"

You can find the complex roots (i.e. zeros) by using the quadratic formula
or you can tweak your equation (instead of adding 15 at the end, use a negative number... that has the effect of lowering the whole shape downward)

Answer 9

ok can we use -15

Answer 10

yes, plot the new equation and see what you get.

Answer 11

https://www.desmos.com/calculator/abydbjucdf

Answer 12

so the new y-int in -15

Answer 13

yes. You now have zeros, but they are not "nice" whole numbers.
I would shift the curve up by 7 (I think). i.e. change the -15 to -8
and see what that looks like

Answer 14

https://www.desmos.com/calculator/rnhnrvm29p

Answer 15

so the new y-int is -8 and your zeros are -3 and 1

Answer 16

ok, that is almost it. shift it down 1 (make the -8 a -9

Answer 17

yes, I am trying to make the zeros -3 and 1. almost got it.

Answer 18

https://www.desmos.com/calculator/psh5hg83x9

Answer 19

So would the answer be the parabola is going up because it is positive. And the y-int is -9 while the zeros are -3 and 1.

Answer 20

yes, if you "zoom in" you can see we go nice zeros, and a nice y-intercept
(and a nice vertex, if you need that )

Answer 21

the parabola is going up because the "a" coefficient on the x^2 term is positive.

Answer 22

we say it is "convex" ( it's easier to remember "holds water" like a cup)

Answer 23

if you know how to factor your equation you will find the zeros will be -3 and 1
you can use the equation to find its vertex is -b/(2a) or -6/(2*3) = -6/6= -1
i.e. the x value of the vertex is -1 (which we see from the graph, it is)

Answer 24

ok thank you so much I have one more question could you please help me?

Answer 25

It is basically about what you just said about the vertex

Answer 26

?

Answer 27

do you want me to open a new question

Answer 28

no. what is the question ?

Answer 29

5. The safety inspector notes that Ray also needs to plan for a vertical ladder through the center of the coaster's parabolic shape for access to the coaster to perform safety repairs. Find the vertex and the equation for the axis of symmetry of the parabola, showing your work, so Ray can include it in his coaster plan.

Answer 30

Find the vertex and the equation for the axis of symmetry

if you have an equation
y = a x^2 + b x + c
(and you do!)
the x value of the vertex is at x= -b/(2a)
can you do that calculation ? (I hope so, I posted it up above)

Answer 31

yes you would get -6/6=-1

Answer 32

ok, (but if you write this up, you should explain a is 3 and b is 6 )

the "axis of symmetry" is a line that divides the parabola in half.
it is
x=-1

so you have the axis of symmetry and the x value for the vertex.
that last thing to find is the value of the vertex
use your equation to find y when x is -1
can you do that?
(btw, you know the answer by looking at your graph)

Answer 33

so the axis of symmetry is the same as the x value of the vertex

Answer 34

sort of. the answer is
x= -1
but (unfortunately) that is a bit confusing. If we say that is the *equation of a line*
x=-1 means a line where all the points on the line have an x value of -1 and the y value can be any number.

if you did not know it was a line, then in algebra, x=-1 would mean the x variable has a value of -1, which is a different idea,

so you should say:
the axis of symmetry is the line x= -1
(then the reader will know that x= -1 is the equation of a line)

Answer 35

but, can you find the y value of the vertex ? (using x=-1 in your equation for the parabola)

Answer 36

https://www.desmos.com/calculator/os3npoyhof

Answer 37

so the y value is 0

Answer 38

yes, one of the zeros is +1
which means y=0 when x is 1

but you want to use x= -1

Answer 39

ok so it is -12

Answer 40

and you might want to practice doing it by hand, because they will expect you to know how.

yes, the vertex is at (-1, -12)
can you find that point on your graph of the parabola?

Answer 41

https://www.desmos.com/calculator/4dy7eewmec

Answer 42

yes i found the point

Answer 43

Thank you so much not only did you help me get the answer, now I know how to do these types of problems :)