Answers to Tutorials in Equation of Parabola

The anwsers to the tutorial in Equation of Parabola are presented.

Question:
2 - Keep the values of a, h and k as above (do not change the positions of the sliders). Find the equation of the directrix and the coordinates of the vertex V and focus F. Find the equation of the axis of symmetry of the parabola (line through V and F).

Answer:
directrix: vertical (green) line: x = -1
Vertex V: (0 , 0)
Focus F: (1 , 0)
Axis of Parabola line y = 0 (x axis).


Question:
3 - Use the top slider to set a = 2 and answer the same questions as in part 2 above.

Answer:
directrix: vertical (green) line: x = -2
Vertex V: (0 , 0)
Focus F: (2 , 0)
Axis of Parabola line y = 0 (x axis).

Question:
4 - Set a = 1, h = 0 and change k (using the slider). Find a relationship between the y-coordinate of F and parameter k. Find a relationship between the y-coordinate of V and k. Find a relationship between the position (or equation) of the axis of the parabola and k. Does the position of the vertex change?

Answer:
y coordinate of focus F = k
y coordinate of vertex V = k
equation of axis of parabola y = k.
Yes the position of the vertex change as k changes.


Question:
5 - Set a = 1, k = 0 and change h (using the slider). Find a relationship between the x-coordinate of F and parameter h. Find a relationship between the x-coordinate of V and h. Find a relationship between the position (or equation) of the directrix of the parabola and h. Does the position of the axis change?

Answer:
x coordinate of focus F = h
x coordinate of vertex V = h
equation of directrix of parabola x = h - 1.
No, the position of the axis of the parabola does not change as h changes.

Question:
6 - Use parts 1,2,3,4 and 5 above to find the coordinates of V and F and the equations of the directrix and axis of the parabola in terms of h and k.

Answer:
Vertex V is at the point (h , k).
Focus F is at the point (h + 1 , k)
Equation of directrix: x = h - 1
Equation of axis: y = k

Question:
7 - Set a = 1, k = 0 and change h. Which values of h give two y-intercepts? Which values of h give no y-intercepts? Which values of h give one y-intercept?Explain your answers analytically.(Hint: find the y-intercepts by setting x = 0 and solve for y).

Answer:
two y-intercepts when h < 0
one y-intercept when h = 0
no y-intercept when h > 0
analytical
To find the y-intercepts, set x = 0 to zero in the equation (y - k)
2 = 4a(x - h) (coefficient a is equal to 1) of the parabola and solve for y.
(y - k)
2 = -4h
If h < 0, the above equation has two real solutions
y = k + sqrt(-4h) and y = k - sqrt(-4h)
If h = 0, the above equation has one real solution
y = k
If h > 0, the above equation has no real solutions.

Question:
8 - Investigate the x-intercept. Explain why the parabola as defined above has one x-intercept only.

Answer:
To find the x intercept we set y = 0 in the equation (y - k)
2 = 4a(x - h) and solve for x.
k
2 = 4a(x - h)
The above equation will always have one solution given by
x = h + k
2 / 4a
Question:
9 -
Exercise: Show that the following equation
y2 - 4y - 4x = 0

can be written as
(y-k)2 = 4a(x - h)

Hint: put all terms with y and y
2 together in one side and all terms with x in the other side of the equation. Complete the square for the expression containing y and y2.
Find a, h and k. Find the coordinates of V and F. Find the equations of the axis and directrix of this parabola. Put the values of a, h and k in the applet and check your answer.

Answer:
We rewrite the equation as
y
2 - 4y = 4x
complete the square
y
2 - 4y + (-4/2)2 = 4x + (-4/2)2
(y - 2)
2 = 4(x + 1)
hence: a = 1, k = 2 and h = -1
Vertex at V(h , k) = (-1 , 2)
Focus at F(h + a, k) = (0 , 2)
Axis, horizontal line, y = k = 2
Directrix, vertical line, x = h - a = -2