Answer:
[tex]\displaystyle f(x)=\frac{7}{15}(x-1)(x+3)[/tex]
Step-by-step explanation:
Equation of the Parabola
The general form of the equation of the parabola is:
[tex]f(x)=ax^2+bx+c[/tex]
We can try to find the values of a,b, and c by using the three given points (1,0),(-3,0),(-9,28).
However, we'll use an easier method. There is another form of the parabola in case we know its roots, also called zeros or x-intercepts. If p and q are the roots of f, then f can be expressed as:
[tex]f(x)=a(x-p)(x-q)[/tex]
We already know the values of p=1 and q=-3, thus replacing them into the equation, we have:
[tex]f(x)=a(x-1)(x+3)[/tex]
We only need to find the value of a. We do that by using the point (-9,28):
[tex]28=a(-9-1)(-9+3)[/tex]
Operating:
[tex]28=a(-10)(-6)=60a[/tex]
Solving for a:
[tex]\displaystyle a=\frac{28}{60}=\frac{7}{15}[/tex]
Thus, the equation of the parabola is:
[tex]\boxed{\displaystyle f(x)=\frac{7}{15}(x-1)(x+3)}[/tex]
