Write a rule that represents the function

Question

(0,0)(1,4)(2,16)(3,36)(4,64)

Answer 1

(0,0)(1,4)(2,16)(3,36)(4,64)

first, is it obvious that the function will not be a straight line? The y value increases faster than the x, and so some kind of power is involved.

4, 16, 36, and 64 are all square of integers--which are 2, 4, 6, and 8. Note that each of these is 2 X the given x value. Thus, it appears that our function is y = (2*x)² . TEST this by plugging in the various values of x.

Answer 2

If you write the coordinates in order and subtract the next y-value from the previous, you will notice a pattern. The first time you do it, you get 4, 12, 20, and 28. When you subtract these numbers, you get a constant 8 as the difference. When the numbers stop changing, you can stop. Since you did this subtraction step twice, "2" is the highest exponent of the function... A quadratic.

Since all quadratics are in the form y = ax²+ bx + c, we can find a, b, and c by using any 3 points from above.

(0,0) (1,4) (2,16) will be used

0 = a(0)² + b(0) + c

0 = c

4 = a(1)² + b(1) + c

4 = a + b + 0

4 = a + b

16 = a(2)² + b(2) + c

16 = 4a + 2b + 0

16 = 4a + 2b

Since 4 = a + b... b = 4 - a

Then 16 = 4a + 2(4-a)

16 = 4a + 8 - 2a

16 = 2a + 8

8 = 2a

4 = a

Plug a = 4 into a + b = 4

4 + b = 4

b = 0

Since a = 4, b = 0, c = 0

y = 4x² + 0x + 0

y = 4x²

Answer 3

each point is on the parabola y=4x² opening upwards with vertex or minimum point at the origin, (0,0)

each point on the parabola is (x,y) = (x,4x²) = (y^1/2/2,y)

where the distance from every point (x,y) on the parabola, to the parabola's focus, (0,1/16) = distance from (x,y) to the directrix line, y=-1/16 The vertex is midway between the focus and the directrix. Every point on the parabola is equidistant to the focus and directrix.

The point (1,4) is 4 1/16 or 65/16 above the directrix. It is also 4 1/16 or 65/16 from the focus (0,1/16)

The point (2,16) is 16 1/16 or 257/16 above the directrix and also 257/16 from the focus (0,1/16)

the point (3,36) is 36 1/16 above the directrix and also 36 1/16 from the focus

the point (4,64) is 64 1/16 above the directrix and also 64 1/16 from the focus