Find the equation of the parabola that passes through the points:(0,5) (1,7) and (-2,19)

Answer:The equation of the parabola is: [tex]y=3x^2-x+5[/tex] Step-by-step explanation:The standard form of a parabola is given as: [tex] y=ax^2+bx+c[/tex]The three points on the parabola are (0,5), (1,7), and (-2,19).Plug in the three points and find three equations in a,b and c Using point (0,5) in the equation, we get [tex] a(0)^2+b(0)+c=5\\c=5[/tex]Using point (1,7) in the equation, we get[tex] a(1)^2+b(1)+c=7\\a+b+c=7[/tex]Using point (-2,19) in the equation, we get[tex] a(-2)^2+b(-2)+c=19\\4a-2b+c=19[/tex]Plug in the value of c=5 in the last two equations. This gives, [tex]a+b+5=7\\a+b=7-5\\a+b=2----- 4\\\\4a-2b+5=19\\4a-2b=19-5\\4a-2b=14\\2(2a-b)=14\\2a-b=7[/tex]Now, add the two new equations. This gives,[tex]a+b+2a-b=2+7\\ 3a=9\\a=\frac{9}{3}=3[/tex]Now, plug in [tex]a=3[/tex] in equation 4 gives,[tex]3+b=2\\b=2-3=-1[/tex]Therefore, the equation of the parabola is:[tex]y=3x^2-x+5[/tex]