Question

How do you graph `y= 6x^2` and show the vertex, axis of symmetry, and x-intercepts of the equation?

Answer 1

Graph of `y = 6x^2` is available as part of this solution.

`Vertex = (0,0)`

Axis of Symmetry is the line `x = 0`

x-intercept `x =(0,0)`

Explanation:

We are given the quadratic equation `y = 6x^2` `color(red)(..Equation.1)`

Graph of the Quadratic Equation `y = 6x^2` is below:

Let us investigate the graph:

How do we find the X Intercept?

To find the x-intercept of the equation we let `y = 0` and then solve for `x`.

Using Point Notation it is written as `(x, 0)`

How do we find the Y Intercept?

To find the y-intercept of the equation we let `x = 0` and then solve for `y`.

Using Point Notation it is written as `(0, y)`

How do we find the Axis of Symmetry?

x-coordinate of the Vertex is the Equation of the Axis of Symmetry of the Parabola.

`color(green)(Step.1)`

How do we find the Vertex of a Parabola?

Standard Form of a Quadratic Equation is

** `y = f(x) = ax^2+bx+c`

The expression `-b/(2a)` will give us the x-coordinate of the Vertex.

Hence, we get `-(0)/2*(6) = 0`

We can obtain the y-coordinate of `y = 6x^2` by substituting the value of x = 0

Hence, `0 = 6(0)^2`

`rArr y = 0`

Hence, our `Vertex = (0,0)`

`color(green)(Step.2)`

x-intercept is obtained by:

Let `y = 0` in `color(red)(..Equation.1)`

`rArr 0 = 6x^2`

`rArr x^2 = 0`

`rArr x = 0` (Note that `sqrt(0) = 0)`

Hence, we can write the x-intercept as `(0,0)`

`color(green)(Step.3)`

How do we find the Axis of Symmetry?

x-coordinate of the Vertex is the Equation of the Axis of Symmetry of the Parabola.

Axis of Symmetry is `x= 0`

We can easily verify all the above results in the graph.

I hope you find the graph and associated explanation useful.