Question

How do you use the important points to sketch the graph of `y=x^2+5x+3`?

Answer 1

`A=(-0.7,0)`
`B=(-4.3,0)`
`C=(0,3)`
`D=(-2.5,-3.25)` min

Explanation:

`y=0 => x^2+5x+3=0`
`=>`
`x_(1,2)=(-5+-sqrt(5^2-4*1*3))/(2*1)=(-5+-sqrt13)/2`
`=>`
`x_1=(-5+sqrt13)/2~~-0.7`
`x_2=(-5-sqrt13)/2~~-4.3`
`=>`
`A=(-0.7,0)`
`B=(-4.3,0)`

---

`x=0 => y=0^2+5*0+3=3`
`=>`
`C=(0,3)`

---

MAX or MIN:
1) `a=1>0 =>` y "smiles" `=>` MIN
2) `y'=2x+5`
`=>`
`y'=0 => 2x+5=0 => x=-5/2=-2.5`
`=>`
`y_((-2.5))=(-2.5)^2+5*(-2.5)+3=-3.25`
`=>`
`D=(-2.5,-3.25)` min

graph (in order to check that all match):
graph{x^2+5x+3 [-10, 10, -5, 5]}