Parametric Parabolas

It hasn't even been a week since we dealt with parabolas, and they come again, this time with actual numbers. The question itself isn't hard, but once again, some thinking is required.

A parabola goes through the points A(17, 3) B(34, z) C(50, -30). B is the maximum point of the parabola. Find point B and the parabola function in standard form.

As you know, a parabola can be written in two ways. One is standard form, with the a, b, and c coefficients, and one is the vertex form. To solve this question, we'll use the vertex form. This form says that the equation a parabola that has a vertex (h, k) is:
y = a(x - h)² + k

We know h and k, so let's plug them in:
y = a(x - 34)² + z

We have two other points, so let's use them:
3 = a(17 - 34)² + z
2 = a(50 - 34)² + z

3 = 289a + z
-30 = 256a + z

Subtract the second equation from the first:
33 = 33a
a = 1
3 = 289a + z
3 = 289 + z
z = -286

The function of the parabola is:
y = (x - 34)² - 286

Open parenthese and combine like terms to switch to standard form:
y = x² - 68x + 1156 - 286
y = x² - 68x + 870

Nice, isn't it?
Nadav

nadavs

Parabola Reflections

Today's question is of a type we didn't have for a long time - pure variables, no numbers. It's very interesting, and although it looks impossible at first, one little step makes it very, very easy.

The parabola y = ax² + bx + c has its vertex on (h, k). The parabola is reflected about the line y = k. The new parabola is: y = dx² + ex + f. Find a + b + c + d + e + f.

Looks impossible, doesn't it? You can try doing some complex calculations showing that -b/2a = h (you should know that), then plug it back, and equate it to y. That's hard. However, parabolas have two forms: standard form and vertex form. Guess what - we have the vertex!

y = a(x - h)² + k

Now we can open the parentheses and find exactly a, b, and c:
y = ax² - 2axh + ah² + k

That means:
a = a
b = -2ah
c = ah² + k

When the line is reflected about y = k, it changes the coordinates. Since the vertex is on y = k, the parabola just switches sides. This means a = -a and b becomes the opposite too (2ah) to keep the vertex.

Notice that the vertex remains the same because the line was reflected about y = k. When you reflect point (x, n) about the line y = z, you end up with (x, 2z - n). If you reflect (h, k) about y = k, you get (h, 2k - k), which is (h, k).

Since each point turns into (x, 2k - y), the new y-intercept, f, becomes (0, 2k - (ah² + k)), which is (0, 2k - ah² - k), or (0, k - ah²).

a and d are opposites, so a + d = 0. b and e behave the same. The only addends left are c and f:
c + f = ah² + k + k - ah² = 2k

This big sum, a + b + c + d + e + f, equals to 2k.

Hope you liked it,
Nadav

nadavs

Inverse Quadratic Function

Today I have another question in the genre of "short question, long answer". Today's question deals with inverse functions, more accurately the inverse of a quadratic function (or more accurately, what becomes a quadratic function). This is not some simple quadratic function, but rather something more difficult to solve.

Find the inverse of y = 2x + sqrt(x)

Short, isn't it? However, the solution isn't that short.

To find the inverse, we need to substitute x and y and then solve for y. Let's do it:
x = 2y + sqrt(y)

To eliminate the square root, move the 2y to the right side and square:
x - 2y = sqrt(y)
x² - 4xy + 4y² = y

Subtract y from both sides and use the distributive property:
x² - 4xy - y + 4y² = 0
x² - y(4x + 1) + 4y² = 0

This calls for the friend of any math student: the quadratic formula. Notice that squaring may add a solution, but there is only one inverse function, so we'll need to eliminate one:
y = ((4x + 1) ± sqrt((4x + 1)² - 16x²)) / 8

Yes, not a pleasant look. Let's work with that a little so see if it gets any better:
y = ((4x + 1) ± sqrt(16x² + 8x + 1 - 16x²)) / 8
y = ((4x + 1) ± sqrt(8x + 1)) / 8

That's the best it gets, but there's still a problem: there are two possible inverse functions here, but only one is right. To find out which one is right, remember that we reached the following conclusion:
x - 2y = sqrt(y)

This means x - 2y must be non-negative. Let's plug the two possible y values to see what we get:
x - ((4x + 1) + sqrt(8x + 1)) / 4 = 4x / 4 - ((4x + 1) + sqrt(8x + 1)) / 4
= (4x - ((4x + 1) + sqrt(8x + 1))) / 4 = (-1 - sqrt(8x + 1)) / 4
In one word: negative. Not good.

Let's plug the other one now to see if it works:
x - ((4x + 1) - sqrt(8x + 1)) / 4 = 4x / 4 - ((4x + 1) - sqrt(8x + 1)) / 4
= (4x - ((4x + 1) - sqrt(8x + 1))) / 4 = (-1 + sqrt(8x + 1)) / 4

As you can clearly see, sqrt(8x + 1) > 0 if x > 0, just like the domain of the original function. That means the inverse of the original function is:
y = ((4x + 1) - sqrt(8x + 1)) / 8

Here is a little confirmation test:
Plug 9 for x in the original function. You get y = 2*9 + sqrt(9) = 21. Now plug 21 in the inverse function and see if it gives back 9:
y = ((4 * 21 + 1) - sqrt(8 * 21 + 1)) / 8
y = (85 - sqrt(168 + 1)) / 8
y = (85 - sqrt(169)) / 8
y = (85 - 13) / 8
y = 72 / 8 = 9

Yes, it is the inverse.

You can find many more interesting math tricks like that on super math tips.

Hope you liked it.
Nadav

nadavs