When discussing a parabola, some definitions are in order. These are:
Parabola: this is the locus of all points (or, set of all points) whose distances from a fixed point (focus) equal their distances from a fixed line (the directrix).
Locus: the set of all points equidistant from both the directrix and the focus.
Equidistant: the same distance from something.
Directrix: the fixed line in the construction of a parabola.
Focus: the fixed point in the construction of a parabola.
Vertex: a point constructed via a perpendicular line from the focus directly to the directrix. It is the point of intersection of the locus and perpendicular line. It is also considered as an axis of symmetry.
To find an equation for a parabola, the distance (the perpendicular dashed line) from a point to a line in the coordinate plane must be calculated. Notice that the distance d from (x,y) (the focus) to the line y = k (directrix) is the distance from (x,y) to (x,k):
The previous idea is used with the following example: Find an equation for the parabola with focus (0,5) and directrix y = -5:
We need to check that the equation is equidistant from the focus and the directrix:
In the previous example, if (0,5) were replaced by (0,1/4) and y = -5 by y = -1/4, the equation for the parabola would be y = x^2. If (0,5) is replaced by (0,1/4a) and y = -5 by y = -1/4a, then the parabola has equation y = ax^2. This demonstrates the theorem that:
When a < 0, the parabola opens down. In this case the directrix is above the x-axis and the focus is below the x-axis. Here is a picture:
A translation is when the parabola moves or 'translates' to the left of the origin, right of the origin, up from the origin or down from the origin. First, we'll consider translations that move to the left and to the right from the origin.
The equation x^2 = 9 has two solutions, 3 and -3. Now consider the related equation
To solve this equaiton, a first idea might be to expand (x - 8)^2, but that is the hard way. The easier way is to take the square roots of each side. You have four possibilities: x - 8 = 3, x - 8 = -3, -(x - 8) = 3 or -(x - 8) = -3. Two of the pairs are equivalent; so use the first two. To solve x - 8 = 3 and x - 8 = -3, add 8 to each side.
The solutions to (x - 8)^2 = 9 are larger by 8 than the solutions to x^2 = 9. This is to compensate for subtracting 8 in that equation. when the solutions are graphed on the number line, they are 8 units to the right. In other words, the solutions to (x - 8)^2 = 9 are the images of the solutions to x^2 = 9 under a translation of 8 units to the right. Look at the two graphs of parabola's y = x^2 and y = (x - 8)^2 (don't be confused about the r(x) expression in the picture below; you can use any letter); Remember that the function (x - 8)^2 has the ordered pair of (8,0) as its vertex:
Note that the number 8 is positive when you ignore the subtraction sign. Also note that the graph of x^2 moved to the right 8 units with a positive 8. If the number were positive 2, the graph would move to the right 2 units. Let's call the number 8 the letter h. Now we have the expression (x - h)^2. When h is positive the graph moves to the right h units. When h is negative you have (x + h)^2 and the graph moves to the LEFT h units. Another way to view it is when h is negative, the graph moves to the left equal to the number -h on the number line (the x-axis):
When h is positive then the x value of the ordered pair is +h; if h is negative, the x value of the ordered pair is -h: (+h, 0) or (-h, 0). In the three graphs above, the h values in their ordered pairs are: (0,0), (8,0), and (-2,0). These are the ordered pairs of the vertices of the three graphs. Also, remember that these various h values are found in the 'x' part of the formula for the graph: for y = x^2 the vertex is (0,0); for y = (x - 8)^2 the vertex is located at (8,0); and for (x + 2)^2 the vertex is located at (-2,0). I highlighted the value for h in each ordered pair. The 'h' values are found within the parentheses involving x and the exponent 2.
So far we've only discussed about left and right translations; that is when x^2 graphs move left and right. But what happens when the graph moves up and down on the y axis? The formula looks like this: x^2 + k, where k is positive. If k is positive, then the graph moves up k units. If the formula looks like x^2 - k, then the graph moves down k units. The plus sign moves it up k units and the minus sign moves it down k units. (Remember, x - -k = x + k). So I keep k positive for both possibilities and look at whether the sign is + or - . Note the following two graphs:
See that the vertex is located at (0,2) for the equation of x^2 +2; and the vertex for the graph of x^2 -2 is (0,-2). The y value of each ordered pair is the same as the last part of the equations. When the equation is a +2, the graph moves up 2 units; when the equation is say a -2, the graph moves down 2 units. Again, the letter k is used in the general equation and it also relates to the y value of the ordered pair for the vertex.
Translations can be done both vertically and horizontally at the same time.
Now we come to the Graph-Translation Theorem and a corollary:
The Theroem:
In a sentence for a graph, replacing x by x - h and y by y - k causes the graph to undergo the translation Th,k*
The Corollary:
The image of the parabola y = ax^2 under the translation Th,k is y - k = a(x - h)^2.
The Graph-Translation Theorem and corollary will be addressed in more detail in Lesson 2 Day 2 information/investigation link. You can go there now or try the challenge first:
* This is pronounced "T sub h k".