Answer:
See below for answers and explanations (along with attached graph)
Step-by-step explanation:
It best helps to convert the equation into vertex form because it provides a lot of information about the characteristics of the parabola:
[tex]f(x)=2x^2-8x+4\\\\f(x)=2(x^2-4x+2)\\\\f(x)+2(2)=2(x^2-4x+2+2)\\\\f(x)+4=2(x^2-4x+4)\\\\f(x)+4=2(x-2)^2\\\\f(x)=2(x-2)^2-4[/tex]
Now, recall that vertex form is [tex]y=a(x-h)^2+k[/tex] where [tex](h,k)[/tex] is the vertex, the axis of symmetry is the line [tex]x=h[/tex], and [tex]a[/tex] is the growth or shrink factor.
Vertex
This is pretty simple as you can just look at the equation, so [tex](h,k)\rightarrow(2,-4)[/tex]
Axis of Symmetry
Again, also simple, so [tex]x=2[/tex] would be the line.
X-intercepts
The beauty of having converted the equation to vertex form is that when we set [tex]f(x)=0[/tex], we can easily find our x-intercepts with minimal work:
[tex]f(x)=2(x-2)^2-4\\\\0=2(x-2)^2-4\\\\4=2(x-2)^2\\\\2=(x-2)^2\\\\\pm\sqrt{2}=x-2\\\\x=2\pm\sqrt{2}[/tex]
So, our x-intercepts are at the points [tex](2+\sqrt{2},0)[/tex] and [tex](2-\sqrt{2},0)[/tex].
Y-intercept
This is just a simple substitution of [tex]x=0[/tex] and everything works out nicely:
[tex]f(x)=2(x-2)^2-4\\\\f(0)=2(0-2)^2-4\\\\f(0)=2(-2)^2-4\\\\f(0)=2(4)-4\\\\f(0)=8-4\\\\f(0)=4[/tex]
So, our y-intercept would be at the point [tex](0,4)[/tex].
Domain
If you try any real number for [tex]x[/tex], there will always be a value for [tex]f(x)[/tex], so the domain of the function is [tex](-\infty,\infty)[/tex] in interval notation.
Range
Recall back to when we found our vertex of [tex](2,-4)[/tex]. Because the leading coefficient of the function is positive, our vertex is the minimum, which means [tex]f(x)[/tex] cannot be lower than -4, but anything higher works, so our range is [tex][-4,\infty)[/tex] in interval notation.
End Behavior
We can easily see that if we plug in a large value for [tex]x[/tex], then [tex]f(x)[/tex] will also be a large value, so our end behavior for the function is that as [tex]x\rightarrow\infty[/tex], then [tex]f(x)\rightarrow \infty[/tex]. Another good indicator of this end behavior is the even leading degree of 2.
I hope these explanations helped! Please feel free to view the attached graph below to help give you a visual!
