Further problems about functions

Summary 1: Function types

  • f(x)=ax+bf(x)=ax+b linear function

  • f(x)=ax2+bx+cf(x)=ax^2+bx+c standard form of quadratic function f(x)=A(x−v)2+Bf(x)=A(x-v)^2+B vertex form of quadratic function

  • f(x)=A(x−v)p+Bf(x)=A(x-v)^p+B Power function

  • f(x)=anxn+an−1xn−1+...+a1x1+a0f(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_1 x^1+ a_0 Polynomial of degree nn.

  • f(x)=Ab(x−v)/uf(x)=A b^{(x-v)/u} exponential function

  • f(x)=log⁡b(x)f(x)=\log_b(x) logarithm

  • f(x)=Asin⁡(u(x−v))+Bf(x)=A\sin(u(x-v))+B sine function

    f(x)=Acos⁡(u(x−v))+Bf(x)=A\cos(u(x-v))+B cosine function

    f(x)=Atan⁡(u(x−v))+Bf(x)=A\tan(u(x-v))+B tangent function

    These are the trigonometric functions.

Exercise 1
A1

  1. Determine the function equation of the straight line ff passing through the points A(2∣3)A(2\vert 3) and B(6∣−2)B(6\vert -2).

  2. Draw the function f(x)=2x2+8x+5f(x)=2x^2+8x+5, and determine the vertex form. Then determine the transformations to get from the graph x2x^2 to the graph ff.

  3. Determine without a calculator the following values:

    (a)144(b)log⁡3(9)(c)sin⁡(π/2)(d)82/3(e)ln⁡(e100)(f)8−1/3(g)cos⁡(π/4)\begin{array}{lll} (a) & \sqrt{144}\\ (b) & \log_3(9)\\ (c) & \sin(\pi/2)\\ (d) & 8^{2/3}\\ (e) & \ln(e^{100})\\ (f) & 8^{-1/3}\\ (g) & \cos(\pi/4)\\ \end{array}
A2

On day 11 it has 100100 bacteria in a dish. How many bacteria does it have on day xx if.

  1. the number of bacteria increases every 33 days by 1.5%1.5\%.
  2. the number of bacteria increases by 1.51.5 every 33 days.
A3

Sketch the graph of the function ff without a calculator. At least 3 points must be drawn exactly.

  1. f(x)=log⁡10(x)f(x)=\log_{10}(x)
  2. f(x)=0.5⋅2xf(x)=0.5\cdot 2^x
  3. f(x)=3−xf(x)=3^{-x}
  4. f(x)=2x+1+1f(x)=2\sqrt{x+1}+1
  5. f(x)=2sin⁡(x−π4)f(x)=2\sin(x-\frac{\pi}{4})
  6. f(x)=2xf(x)=\frac{2}{x}
  7. f(x)=cos⁡(π2x)f(x)=\cos(\frac{\pi}{2}x)
A4

Match the graphs below to the correct functions and give a short reason for each. There is no graph for two functions.

f(x)=−0.5x+1g(x)=−2x2h(x)=x2−x−1i(x)=−3x6+6x5−2x2+1k(x)=x3−x2+2.5l(x)=1m(x)=−x5+2x2n(x)=x6+x4\begin{array}{lll} f(x) = -0.5x+1\\ g(x) = -2x^2\\ h(x) = x^2-x-1\\ i(x) = -3x^6 + 6x^5 -2x^2 +1\\ k(x) = x^3 - x^2 + 2.5\\ l(x)=1\\ m(x) = -x^5 + 2x^2\\ n(x) = x^6 + x^4\\ \end{array}
A5

One of the classical problems of the antiquity was the so-called cube doubling: To a cube another cube with double volume is to be constructed. Find the ratio of the edge lengths or the surface areas of the two cubes.

A6 (Sierpinski Carpet)

The Sierpinski triangle is a structure that is created iteratively: From an equilateral triangle, the middle triangle is removed first. From the remaining three triangles the same the middle triangle is removed again, and so on. It is interesting to see how the total area and perimeter of all black triangles change: The area decreases after each step, the perimeter increases.

  1. Determine the area and perimeter of the Sierpinski triangle after 1, 2, 3, 4, ... steps in relation to the initial triangle with side length 11.
  2. Prepare a table for this purpose. What is the area and the perimeter of the Sierpinski triangle if you continue the process indefinitely?