The equation of the parabola is given is the Vertex Form. The general form of a quadratic equation in Vertex Form is:
[tex]y=a(x+b)^{2}+c[/tex]
So for our equation, [tex]a=1, b=3,[/tex] and [tex]c=-4[/tex]. Now let's solve the question.
1. Line of Symmetry:
Line of symmetry is given as [tex]x=-b[/tex], so our line of symmetry is [tex]x=-3[/tex].
2. Vertex:
Vertex is given as [tex](-b,c)[/tex], so our vertex is [tex](-3,-4)[/tex].
3. Roots:
We find the roots by setting [tex]y=0[/tex]. Thus, we have
[tex]0=(x+3)^{2} -4\\4=(x+3)^{2}\\[/tex]
So,
[tex]x+3=2[/tex] and [tex]x+3=-2[/tex]
So, solving these 2 equations we have [tex]x=-1, -5[/tex]
4. Y-Intercept:
To find y-intercept, we set [tex]x=0[/tex]. So we have
[tex]y=(0+3)^{2}-4\\y=5[/tex]
5. Minimum/Maximum:
A quadratic equation has minimum if [tex]a[/tex] is positive and maximum is [tex]a[/tex] is negative. Hence, this function has a minimum since [tex]a[/tex] is positive. The value of the minimum is [tex]y=c[/tex]. So for our question, the minimum is [tex]y=-4[/tex].
ANSWERS:
1. Line of Symmetry: [tex]x=-3[/tex]
2. Vertex: [tex](-3,-4)[/tex]
3. Roots: [tex]x=-1, -5[/tex]
4. y-intercept: [tex]y=5[/tex]
5. Minimum Value: [tex]y=-4[/tex]
