How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #y = 2x^2 - 6x - 36#?

Question

Kennedy Adams · Accepted Answer

The function has a minimum at #x=1.5#

Axis of symmetry #x=1.5#

Given -

#y=2x^2-6x-36#
#dy/dx=4x-6#
#(d^2y)/(dx^2)=4>0#
#dy/dx=0 =>4x-6=0#
#x=6/4=1.5#

At #x=1.5; dy/dx=0;(d^2y)/(dx^2)>0#

The function has a minimum at #x=1.5#

Axis of symmetry #x=1.5#

Graph -

Julia Adams · Answer

undefined To find the axis of symmetry of a quadratic function $ y = ax^2 + bx + c $, use the formula $ x = -\frac{b}{2a} $.

For the function $ y = 2x^2 - 6x - 36 $, the axis of symmetry is $ x = \frac{-(-6)}{2 \cdot 2} = \frac{6}{4} = \frac{3}{2} $.

To find the maximum or minimum value, substitute the axis of symmetry into the function to find the corresponding y-value.

$ y = 2\left(\frac{3}{2}\right)^2 - 6\left(\frac{3}{2}\right) - 36 $

$ y = 2 \cdot \frac{9}{4} - 9 - 36 $

$ y = \frac{18}{4} - \frac{36}{4} $

$ y = \frac{-18}{4} $

$ y = -\frac{9}{2} $

So, the axis of symmetry is $ x = \frac{3}{2} $, and the minimum value of the function is $ y = -\frac{9}{2} $.

To graph the function, plot the vertex (axis of symmetry) and two additional points, then draw a parabola through those points.