The corollary from the previous lesson states that the image of the parabola y = ax^2 under the translation Th,k* is y - k = a(x - h)^2. The general equation, y - k = a(x - h)^2, has a name. It is called the vertex form of an equation for a parabola.
So, when you are told to sketch a parabola given an equation like y = 3(x - 6)^2 + 7, the first step is to put the equation into 'vertex form'. You would write: y - 7 = 3(x - 6)^2. When the equation is put into the vertex form, then k and h are easily found; here, k = 7 and h = 6.
This means that the graph is 6 units to the right and 7 units above y = 3x^2. The graph is parabola with vertex (6, 7). Since the '3' in 3x^2 is positive, the graph opens upward. To find some other points start with the vertex and use symmetry. The axis of symmetry is a vertical or horizontal line (vertical in this case) that runs through the vertex and through the directrix. See Lesson 2 Day 1 information/ investigation for a definition of the axis of symmetry.
The x-value 1 unit to the left of the axis of symmetry is x = 5. Substitute x = 5 into the formula y - 7 = 3(x - 6)^2: y - 7 = 3(5 - 6)^2 -> y - 7 = 3 -> y = 10, so the ordered pair is (5, 10). Then by symmetry a point 1 unit to the right of the axis of symmetry has the ordered pair (7, 10). In a similar manner you can verify that (4, 19) and (8, 19) are on the parabola. See the graph below:
Lets discuss the properties of the variable 'a' in the formula y - k = a(x - h)^2. When a is positive, the parabola opens upward. If a is negative, the parabola opens downward. When a = 1 or -1 the 'arms' are considered to be the standard distance apart. When a > 1 or < -1 the 'arms' move closer together. When -1 < a < 1 the 'arms' move farther apart. If you observe the graph that opens up and has the widest arms = 1/2 x^2. The graph with the narrowest arms (opening upward) is the 10x^2 graph. The one in the middle is the 1x^2 graph. The same idea holds for negative values for a:
The parabola's we've looked at all either open up or down even with the various translations. But what does the equation look like for a parabola that opens to the left or right?
Remember that the equation we've been dealing with is y - k = a(x - h)^2. The key is when the expression being squared (here the (x - h)) changes: a(y - k)^2 = x - h. Say we have x = y^2; note that when you take the square root of both sides you have y = +/- sqrt(x). (Note that +sqrt(x) is the upper graph and -sqrt(x) is the lower graph):
In the previous graph, a = +1. The general equation is 1(y - 0)^2 = (x - 0) or to put it more simply y^2 = x or x = y^2. What if a = -1? When a = -1 the equation is x = -(y^2) and thus, you have:
Do you think the Graph-Translation Theorem holds for parabolas that open to the side?
If yes, how do you know?
Now lets look at some more investigations. I would advise graphing all of these parabola's for practice:
Suppose the parabola with equation y = 2x^2 undergoes the translation T[2,-3]. Find an equation for its image.
Vertex form is :y - k = a(x - h)^2. Also given is T[h,k], with h = 2 and k = -3. Also given is that a = 2 in the expression 2x^2. Now fill in a, h and k with the given numbers:
y + 3 = 2(x - 2)^2 -> and there's the answer.
The next investigation given is a parabola is congruent to y = 3x^2 and has a vertex of (-2, 2). What is the equation for it?
In case you didn't know, translations can be done horizontally and vertically at the same time. a translation Th,k, slides a figure h units to the right or left depending on the sign and k units up or down again, depending on the sign of the number (+ or -). The figure slides such that the vertex with ordered pair (x,y) corresponds to (h,k). Think of x corresponding to h and y with k from the vertex form of the equation for a parabola. So this equation has h = -2, k = 2 and a = 3 from 3x^2. The equation is: y - 2 = 3(x + 2)^2.
A parabola with vertex (2, -5) opens down. If the parabola is congruent to y = 7x^2, what is the equation for the parabola?
Since the parabola is congruent to 7x^2 and opens down, that means that a = -7. Opening down gives the negative sign and the 7 comes from the congruency with 7x^2. So we have -7x^2 so far. The information from the previous problem tells us that h = 2 and k = -5 therefore, the equation is: y + 5 = -7(x - 2)^2
Now lets try an investigation using x = y^2 as the beginning parabola. See the picture above for this graph. It opens to the right. What is the equation for this parabola with T3,-1? Remember, here a = 1, h = 3 and k = -1 but which expression carries the exponent 2 and which expression has the a in front of it?
You have y + 1 and x - 3 and then remembering that a and the exponent corresponds with the expression with y in it you have:
* Th,k is pronounced T sub h,k.