# 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. k2 = 4a(x - h) The above equation will always have one solution given by x = h + k2 / 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 y2 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 y2 - 4y = 4x complete the square y2 - 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