# determine constants

#### euclid2

Determine values of a,b,c such that the graph $$\displaystyle y=ax^2+bx+c$$ has a relative maximum at $$\displaystyle (3,12)$$ and crosses the y axis at (0,1)

#### skeeter

MHF Helper
Determine values of a,b,c such that the graph $$\displaystyle y=ax^2+bx+c$$ has a relative maximum at $$\displaystyle (3,12)$$ and crosses the y axis at (0,1)
first, you have two points on the curve given to you ...

(0,1) ... $$\displaystyle 1 = a(0^2) + b(0) + c$$ ... now you know $$\displaystyle c = 1$$

(3,12) ... $$\displaystyle 12 = a(3^2) + b(3) + 1$$

this gives you the equation $$\displaystyle 9a + 3b = 12$$ , or simplified, $$\displaystyle 3a + b = 4$$

now you need another equation ... relative extrema occur at critical values; in this case where $$\displaystyle y' = 0$$.

$$\displaystyle y = ax^2 + bx + 1$$

$$\displaystyle y' = 2ax + b$$

$$\displaystyle 2ax + b = 0$$ at $$\displaystyle x = 3$$ ...

$$\displaystyle 6a + b = 0$$

there is your second equation in terms of $$\displaystyle a$$ and $$\displaystyle b$$ ... solve the system.

• euclid2