Respuesta :
Answer:
1) [tex]x=-4[/tex] [tex]D=(-\infty,\infty)[/tex] [tex]R=[-9,\infty)[/tex] 1st graph below x-intercepts S={-7,-1}
2) Opens Down Vertex: (-2,4) Axis of Symmetry [tex]x=-2[/tex] x-intercept [tex]S=\left \{ 0,-4 \right \} y-intercept: (0,0)[/tex]
B Domain: [tex]D=(-\infty,\infty)[/tex]
Range[tex]R=(-\infty,4][/tex]
C This function increases from [tex](-\infty,-2)[/tex] And decreases from [tex](-2,\infty)[/tex]
Step-by-step explanation:
1) The vertex of the parabola is found when we rewrite the common formula:
[tex]f(x)=ax^{2}+bx+c[/tex]
Into this way:
[tex]f(x)=a(x-h)^{2}+k[/tex] The Vertex is found by:
[tex]\\h=\frac{-b}{2a};k=\frac{-\Delta }{4a}\\f(x)=y=x^{2}+8x+16-9\Rightarrow y=x^{2}+8x+7\Rightarrow h=\frac{-8}{2}\Rightarrow h=-4\Rightarrow k=\frac{-\Delta }{4a}\Rightarrow \:k=-\frac{8^{2}-(4*1*7)}{4*1}\Rightarrow k=-9\\ (h,k)\Rightarrow (-4,-9)[/tex]
That's why we could rewrite the trinomial as this:
[tex]f(x)=(x+4)^2-9\\[/tex]
Give the equation of the parabola's axis of symmetry, this is given by tracing a vertical line through the parabola vertex.
[tex]x=-4[/tex]
The intercepts are the roots/zeros:
[tex]y=x^{2}+8x+7\Rightarrow y=(x+7)(x+1) \Rightarrow x'=-7,\:x''=-1 \:S=\left \{ -7,-1 \right \}[/tex]
Domain:
Since the function has no restrictions therefore it is continuous and defined for any value of x ∈ Real Set
[tex]D=(-\infty,\infty)[/tex]
Range
As the minimum point -9 is lowest y-coordinate the Range includes this value up to infinite values
[tex]R=[-9,\infty)[/tex]
(First Graph)
2) [tex]f(x)=-x^{2}-4x[/tex]
As the parameter a <0 then the graph opens down.
Vertex:
[tex]h=\frac{-b}{2a};k=\frac{-\Delta }{4a}\Rightarrow h=-\left ( \frac{-4}{2(-1)} \right ); k=-\left ( \frac{(-4)^{2}-4(-1)(0)}{4(-1)} \right )\Rightarrow (-2,4)[/tex]
Axis of Symmetry
[tex]x=-2[/tex]
x-intercept
[tex]\\y=-x^{2}-4x\Rightarrow 0=-x^{2}-4x\Rightarrow x^{2}+4x=0\Rightarrow x(x+4)=0\Rightarrow S=\left \{ 0,-4 \right \}[/tex]
y-intercept
c=0 then (0,0).
B Domain:
Similarly, since the function has no restrictions therefore it is continuous it is defined for any value of x ∈ Real Set
[tex]D=(-\infty,\infty)[/tex]
Range
[tex]R=(-\infty,4][/tex]
C
This function increases from [tex](-\infty,-2)[/tex] Or we can represent this interval like this:
And decreases from [tex](-2,\infty)[/tex]

