Answer:
1) [tex]f(x)=(x+1)(x-3)[/tex]
2) [tex]f(x)=(x-1)^2-4[/tex]
3) [tex]f(x)=x^2-2x-3[/tex]
Step-by-step explanation:
So we have a graph and we know that its roots are at x=-1 and x=3.
We also know the vertex is at (1,-4). With that, we can figure out the three forms.
Factored Form:
The factored form, as given, is:
[tex]f(x)=a(x-r_1)(x-r_2)[/tex]
We already know the roots of -1 and 3. So, substitute:
[tex]f(x)=a(x-(-1))(x-3)[/tex]
Simplify:
[tex]f(x)=a(x+1)(x-3)[/tex]
Now, we just need to figure out a. To do so, we can use the vertex. Since the vertex is at (1,-4), this means that f(1) is -4. So, substitute 1 for x and substitute -4 for f(x):
[tex]-4=a(1+1)(1-3)[/tex]
Add and subtract:
[tex]-4=a(2)(-2)[/tex]
Multiply:
[tex]-4=-4a[/tex]
Divide both sides by -4:
[tex]a=1[/tex]
So, our factored form is:
[tex]f(x)=(x+1)(x-3)[/tex]
Vertex Form:
The vertex form is:
[tex]f(x)=a(x-h)^2+k[/tex]
Where (h,k) is the vertex.
We already know the vertex is (1,-4), so substitute 1 for h and -4 for k.
We also previously determined that a is 1, so substitute that also. So:
[tex]f(x)=(1)(x-(1))^2+(-4)[/tex]
Simplify:
[tex]f(x)=(x-1)^2-4[/tex]
Standard Form:
To acquire the standard form, simply expand the factored or vertex form. I'm going to expand the factored form:
[tex]f(x)=(x+1)(x-3)[/tex]
FOIL:
[tex]f(x)=x^2-3x+x-3[/tex]
Combine like terms:
[tex]f(x)=x^2-2x-3[/tex]
And we're done!
