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

1 Answer
Jan 3, 2018

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]}