Question

Find the vertex, axis of symmetry, domain, and range of the graph of y = −2x2 − 6x + 1. 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 = −2x2 − 6x + 1.

Answer 1

so, for the equation:
\[-2x^2 - 6x + 1\]
the domain is clearly all real numbers because no value of x is undefined (divide by 0, etc.)

Answer 2

you can find the x value of the vertex by using the equation:
\[\frac{-b}{2a} \]

Answer 3

or -3/2

Answer 4

a quadratic equation is symmetrical around its vertex. Because this equation is in terms of x, the equation for the line of symmetry is x = -3/2

Answer 5

the line is x = -3/2 (the x coordinate of the vertex)

Answer 6

now we need to find the y coordinate of the vertex

Answer 7

by definition, a function takes in an x value and gives you a y value

Answer 8

so if you plug the x coordinate of the vertex into the equation, you get the y coordinate

Answer 9

\[-2(\frac{-3}{2})^2 - 6(\frac{-3}{2}) + 1\]

Answer 10

let me go get my calculator ( I am lazy :P)

Answer 11

I am getting 11/2 as the result

Answer 12

so that is the y coordinate of the vertex

Answer 13

so you know that the vertex is:
\[(\frac{-3}{2},\frac{11}{2})\]

Answer 14

now. because "a" is negative, if you were to graph the equation, it would open downward

Answer 15

that means that the y coordinate of the vertex would be the maximum value