Question

How do you graph and label the vertex and axis of symmetry `y=-4x^2-1`?

Answer 1

Take the graph of `y=x^2`, reflect it over the y-axis, translate the graph 1 unit down, and stretch the graph by a scale factor of 4. Put numbers into the equation and plot them on the graph. You should get this:
graph{-4x^2-1 [-10, 10, -10, 1]}

Explanation:

We can find the vertex algebraically. The x-coordinate of the vertex is found by using `x=-b/(2a)`. However, there is no b-value. Notice how, in the standard form of a quadratic equation (`y=ax^2+bx+c`), no `bx` term exists. When this happens, we replace `b` with `0`. `0` divided by anything is still `0`, so our x-value is `0`.

To find our y-value, we take our x-value and plug it into our original equation:

`y=-4(0)^2-1`
`y=-1`

We now have our vertex, `(0,-1)`. What about our axis of symmetry? Well, the axis passes through the vertex and would have to be a vertical line in our case. Using these, we can say that our axis of symmetry is `x=0`.