Question

How to find range and domain?

Answer 1

Part 1 − Find the vertex, axis of symmetry, domain, and range of the graph of y = −3x2 − 3x + 4. Show all work for full credit.

Part 2 − Using complete sentences, explain how you can determine the axis of symmetry, the domain, and range without graphing y = −3x2 − 3x + 4.

Answer 2

hello again

Answer 3

Domain: All real numbers, Vertex: 3/-6 = -1/2; the formula for finding the vertex is -b/2a

Answer 4

almost all answers come from finding the vertex of \(y = −3x^2 − 3x + 4\)

Answer 5

as @Dahlioz said, it is \(-\frac{b}{2a}\) for the first coordinate, which in this case is \(-\frac{-3}{2\times -3}=-\frac{1}{2}\)
second coordinate is what you get when you replace \(x\) by \(-\frac{1}{2}\)

Answer 6

can you find the second coordinate? you need that one too, for the range

Answer 7

So I'm trying to find the range and domain, step by step.. can you explain it? I don't just want an answer :) x

Answer 8

you need the second coordianate of the vertex to find the rangel
so your first job is to get the first coordinate as \(-\frac{1}{2}\) and your second job is to compute \(-3(-\frac{1}{2})^2-3\times (-\frac{1}{2})+4\)

Answer 9

This equation is in the form ax^2 + bx + c, the formula for finding the vertex is -b/2a, the range of the equation is = all real numbers equal to or greater than the value of −3x2 − 3x + 4 when x is replaced by the "value" of the vertex, in this case -1/2

Answer 10

There are no imaginary or complex numbers involved, so the domain of the function is = all real numbers

Answer 11

this is similar to the last one
you have to compute \[-3(-\frac{1}{2})^2-3\times (-\frac{1}{2})+4\]
\[=-3\times \frac{1}{4}+\frac{3}{2}+4\]
\[=\frac{-3}{4}+\frac{6}{4}+4\]
\[=\frac{3}{4}+4=\frac{19}{4}\]

Answer 12

once you have the vertex at \((-\frac{1}{2},\frac{19}{4})\) you can answer the other questions
lets take them in turn

Answer 13

axis of symmetry:
since the first coordinate of the vertex is \(-\frac{1}{2}\) the "axis of symmetry" is \(x=-\frac{1}{2}\)

Answer 14

domain:
since this is a polynomial, the domain is all real numbers

Answer 15

range :
this is a parabola that opens down because the leading coefficient \(-3\) is negative
since the second coordinate of the vertex is \(\frac{19}{4}\) that is the highest the graph goes, making the range \(y\leq \frac{19}{4}\) or \((-\infty,\frac{19}{4}]\)

Answer 16

as you can see most everything comes by first finding the vertex

Answer 17

Wow, thank you!! I understand this :) I can't assure you that I won't ask another question but I can assure that it won't be like this problem :)

Answer 18

no problem
good luck!

Answer 19

Great explanation @satellite73 , only thing I would like to add is that the leading coefficient is the coefficient of the highest the degree term, in this case x^2, as -3 was the coefficient of both x^2 and x

Answer 20

@Dahlioz Thank you!! :)

Answer 21

No problem :)