How do you find the axis of symmetry, graph and find the maximum or minimum value of the function #y = x^2 - 2x - 15#?

1 Answer
Aug 14, 2017

Refer to the explanation.

Explanation:

#y=x^2-2x-15# is a quadratic equation in standard form:

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

where:

#a=1#, #b=-2#, and #c=-15#.

Vertex

The maximum or minimum point of a parabola. Since #a>0#, the vertex is the minimum and the parabola opens upward.

The vertex form for a quadratic equation is:

#a(x-h)^2+k#,

where:

#h# is the axis of symmetry and #(h,k)# is the vertex.
To determine #h# from the standard form, use the formula: #h=(-b)/(2a)#.

#h=(-(-2))/(2*1)=2/2=color(red)1#

To determine #k# from the standard form, substitute #y# for #k#, and substitute the value of #h# for #x# and solve for #k#.

#k=color(red)(1)^2-2(color(red)(1))-15#

#k=1-2-15#

#k=color(blue)(-16#

The vertex is #(color(red)1,color(blue)(-16))#.

We will also need to find the x-intercepts by substituting #0# for #y# and solving for #x#.

X-Intercepts

The values of #x# when #y=0#.

Factor:

#0=x^2-2x-15#

Find two numbers that when add equal #-2#, and when multiplied equal #-15#. The numbers #3# and #-5# meet the requirements.

#0=(x+color(purple)(3))(xcolor(magenta)(-5))#

#0=(x+color(purple)(3))#

#-color(purple)(3)=x#

Switch sides.

#x=-color(purple)(3)#

#0=(xcolor(magenta)(-5))#

#color(magenta)5=x#

Switch sides.

#x=color(magenta)5#

The x-intercepts are #(-3,0)# and #(5,0)#.

Summary

The axis of symmetry is #color(red)1#.

The vertex is #(color(red)(1),color(blue)(-16))#

The x-intercepts are #(color(purple)(-3),0)# and #(color(magenta)(5),0)#.

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

graph{y=x^2-2x-15 [-15.57, 16.47, -16.77, -0.75]}