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

Jan 3, 2018

Axis of symmetry: $x = 0$

Vertex: $\left(0 , 3\right)$

Y-intercept: $\left(0 , 3\right)$

Other Points: $\left(1 , 4\right)$, $\left(- 1 , 4\right)$, $\left(3 , 12\right)$, and $\left(- 3 , 12\right)$

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:

$y = a {x}^{2} + b x + c$,

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:

$x = \frac{- b}{2 a}$

$x = \frac{0}{2}$

$x = 0$

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: $\left(0 , 3\right)$

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: $\left(1 , 4\right)$

$x = - 1$

$y = - {1}^{2} + 3$

$y = 1 + 3$

Point 2: $\left(- 1 , 4\right)$

$x = 3$

$y = {3}^{2} + 3$

$y = 9 + 3$

$y = 12$

Point 3: $\left(3 , 12\right)$

$x = - 3$

$y = - {3}^{2} + 3$

$y = 9 + 3$

$y = 12$

Point 4: $\left(- 3 , 12\right)$

SUMMARY

Axis of symmetry: $x = 0$

Vertex: $\left(0 , 3\right)$

Y-intercept: $\left(0 , 3\right)$

Other Points: $\left(1 , 4\right)$, $\left(- 1 , 4\right)$, $\left(3 , 12\right)$, and $\left(- 3 , 12\right)$

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]}