Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function `y=x^2 + 3`?

Answer 1

Axis of symmetry: `x=0`

Vertex: `(0,3)`

Y-intercept: `(0,3)`

Other Points: `(1,4)`, `(-1,4)`, `(3,12)`, and `(-3,12)`

Explanation:

In order to graph a parabola, determine the axis of symmetry, vertex, y-intercept, x-intercepts and other points. If the parabola does not cross the x-axis and/or y-axis, you will not have a y-intercept and/or x-intercepts, so you will have to rely on other points.

`y=x^2+3` is a quadratic equation in standard form:

`y=ax^2+bx+c`,

where:

`a=1`, `b=0`, and `c=3`

Axis of symmetry: the vertical line that divides the parabola into two equal halves.

For a quadratic equation in standard form, the formula for the axis of symmetry is:

`x=(-b)/(2a)`

`x=0/2`

`x=0`

Axis of symmetry: `x=0`

Vertex: minimum or maximum point. The `x`-value of the vertex is the axis of symmetry.

To find the `y`-value of the vertex, substitute `0` for `x` and solve for `y`.

`y=x^2+3`

`y=0^2+3`

`y=3`

Vertex: `(0,3)`

Y-intercept: The vertex is also the y-intercept.

Other Points

The parabola does not cross the x-axis, so there are no x-intercepts, but we can find points for values of `x`.

`x=1`

`y=1^2+3`

`y=1+3`

`y=4`

Point 1: `(1,4)`

`x=-1`

`y=-1^2+3`

`y=1+3`

Point 2: `(-1,4)`

`x=3`

`y=3^2+3`

`y=9+3`

`y=12`

Point 3: `(3,12)`

`x=-3`

`y=-3^2+3`

`y=9+3`

`y=12`

Point 4: `(-3,12)`

SUMMARY

Axis of symmetry: `x=0`

Vertex: `(0,3)`

Y-intercept: `(0,3)`

Other Points: `(1,4)`, `(-1,4)`, `(3,12)`, and `(-3,12)`

Plot the points and sketch a parabola through them. Do not connect the dots.

graph{y=x^2+3 [-9.33, 10.67, -2.4, 7.6]}