#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]}
