Question

How do you tell whether the graph opens up or down, find the vertex, and find the axis of symmetry given `y=5x^2+1`?

Answer 1

`y=5x^2+1` opens up, the vertex is (0,1), and the axis of symmetry is on the y axis (equation is x=0)

Explanation:

Up or Down? To tell whether a quadratic equation's graph opens up or down, look at whether the coefficient of x squared (number before x squared) is greater than or less than 0 (it can't be zero, cause then it isn't a quadratic equation). For example, `y=5x^2+1` has 5, so it is positive and opens up. If it is `y=-5x^2+1`, then it has -5, which is negative and opens down.

Vertex? If you are using this structure of quadratic equations (`y=ax^2+bx+c` ) which you are, then you plug b and a of that formula to find the x coordinate of the vertex: `-b/2a`. So since b is 0 (cause your equation has no __x, just x squared and a number 1) and a is 5, you plug it in:
`x=-b/2a`
`x=-(0)/2(5)`
`x=-0/10`
`x=0`

So you have x=0 for the vertex. To find the y coordinate, just plug in x=0 into the equation:
`y=5x^2+1`
`y=5(0)^2+1`
`y=0+1`
`y=1`

So now you have the coordinates for the vertex of `y=5x^2+1`, which is `(0,1)`

Axis of Symmetry? This is easy once you find the vertex. Because the vertex is sitting on the axis of symmetry, you just take the x value for up-down quadratic graphs (not sideways, that would be totally different) which, in this case, is `0`, and set x to always = `0` and there you have it, you got the equation for the axis of symmetry - `x=0`

A really helpful tool to have a visual guide is to use this graphing calculator, desmos.com/calculator where you can plug in any equation you want and have it graph it for you. Good luck on math!