Answer:
[tex]\displaystyle y=\frac{3}{2}x^2-9x+\frac{19}{2}[/tex]
Step-by-step explanation:
Equation of the Quadratic Function
The vertex form of the quadratic function has the following equation:
[tex]y=a(x-h)^2+k[/tex]
Where (h, k) is the vertex of the parabola that results when plotting the function, and a is a coefficient different from zero.
Substituting the coordinates of the vertex (3,-4):
[tex]y=a(x-3)^2-4[/tex]
Now we find the value of a by substituting the point through which the function passes (1,2):
[tex]2=a(1-3)^2-4[/tex]
Operating:
[tex]2=4a-4[/tex]
[tex]4a=6[/tex]
[tex]a=\frac{3}{2}[/tex]
Thus, the equation is:
[tex]\displaystyle y=\frac{3}{2}(x-3)^2-4[/tex]
Expanding the square:
[tex]\displaystyle y=\frac{3}{2}(x^2-6x+9)-4[/tex]
Operating:
[tex]\displaystyle y=\frac{3}{2}x^2-9x+\frac{27}{2}-4[/tex]
Simplifying:
[tex]\boxed{\displaystyle y=\frac{3}{2}x^2-9x+\frac{19}{2}}[/tex]
