You could complete the square to state the vertex.
You could use the quadratic equation to find the roots (which are complex).
Try an example that will require both.
y = x^2 + 2x + 5
Step One
Get the graph. That's included below.
Step Two
Provide the steps for completing the square.
Note: we should get (-1,4)
y= (x^2 +2x ) + 5
y = (x^2 +2x + 1) + 5 - 1
y = (x +1)^2 + 4
The vertex is at (-1,4)
Step Three
Find the roots. Use the quadratic equation. Note that the graph shows us that the equation never crosses or touches the x axis. The roots are complex.
[tex]\text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a} [/tex]
a = 1
b = 2
c = 5
[tex]\text{x = }\dfrac{ -2 \pm \sqrt{2^{2} - 4*1*5 } }{2} [/tex]
[tex]\text{x = }\dfrac{ -2 \pm \sqrt{4 - 20 } }{2} [/tex]
[tex]\text{x = }\dfrac{ -2 \pm \sqrt{-16 } }{2} [/tex]
[tex]\text{x = }\dfrac{ -2 \pm 4i }{2} [/tex]
x = -1 +/- 2i
x1 = -1 + 2i
x2 = -1 - 2i And we are done.
