Just to give the initial idea, this is the code drawing most of the picture in question:
Manipulate[
Show[{
Plot[4 (z - 0.5)^2 - 3, {z, -1, 2}, PlotRange -> {-3.5, 2.2},
Ticks -> None, AxesStyle -> Arrowheads[{-0.03, 0.03}]],
Graphics[{Orange, Dashed, Thickness[0.008], Arrowheads[0.05],
Arrow[{{0.5, -3}, {0.5, 2}}]}],
Graphics[{Text[Style["Axis of symmetry", 12, Orange],
Scaled[{0.7, 0.97}]],
Text[Style["x-intercepts", 12, Red], Scaled[{0.8`, 0.218`}]]}],
Graphics[{Orange, Thickness[0.002],
Arrow[{Scaled[{0.62, 0.934}], Scaled[{0.504`, 0.78}]}]}],
Graphics[{Red, PointSize[0.02], Point[{-0.366, 0}],
Point[{1.366, 0}]}],
Graphics[{Red, Thickness[0.002],
Arrow[{Scaled[{0.8`, 0.25}], Scaled[{0.78`, 0.59`}]}],
Arrow[{Scaled[{0.8`, 0.25}], Scaled[{x, y}]}]}]
}], {x, 0, 1}, {y, 0, 1}]
That is what you should see on the screen:

Now, some explanations.
The main statement is Showwrapping the whole drawing.
The latter consists of the Plot drawing the parabola along with axes specification on one hand and several Graphics statements on the other hand. These draw arrows, text labels and so on.
The whole code is wrapped by the Manipulatestatement. This is technical. Manipulate is used, while the drawing is being done to rapidly find various coordinates. The example is given directly in the code, where I specified the end coordinates of the second red arrow by the dynamically varying coordinates x and y. One finds x and y by moving the sliders, and copy-pastes the corresponding values into the code. Just play with them.
By the same procedure it is easy to draw other elements of your target picture.
After the drawing is finalized, remove the manipulate statement. Done.
The last thing, is that if you need to have bent arrows indicating the characteristic points, you might want to use Bezier curves instead of arrows.
Have fun!
gra=modifiedGraphic
– elbOlita Jul 02 '15 at 18:09File | Save Selection As ...thenImport[<file pathname>]UseInsert | File Path...to get<file pathname>– Bob Hanlon Jul 02 '15 at 18:45Importis just to demonstrate that the plot is saved with the annotations. Pick whatever format (.jpg, .png, .pdf, ...) gives you the best results for your purpose. If you want the notebook to be able to regenerate the annotated plot, add the annotations withEpilogusing graphic primitives. – Bob Hanlon Jul 02 '15 at 19:12