Graphs of Power, Exponential, Logarithmic, and Circular Functions

Power Functions: $y = x^n$, $n \in Q$

Power functions have the general form $y = x^n$, where $n$ is a rational number. The shape of the graph depends heavily on the value of $n$.

• Integer Values of n

  • $n > 0$: The graph passes through (0,0) and (1,1).
  • $n$ is even: The graph is symmetrical about the y-axis (even function).
  • $n$ is odd: The graph is symmetrical about the origin (odd function).
  • $n < 0$: The graph has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.

• Rational Values of n

  • $n = 1/2$: $y = x^{1/2} = \sqrt{x}$. Defined for $x \geq 0$.
  • $n = 1/3$: $y = x^{1/3} = \sqrt[3]{x}$. Defined for all real numbers.

Exponential Functions: $y = a^x$, $a \in R^+$

Exponential functions have the general form $y = a^x$, where $a$ is a positive real number not equal to 1 ($a \in R^+ \setminus {1}$). A crucial case is $y = e^x$.

• Key Features

  • The graph always passes through the point (0, 1) because $a^0 = 1$.
  • If $a > 1$, the function is increasing (exponential growth).
  • If \$0 < a < 1$, the function is decreasing (exponential decay).
  • The x-axis ($y = 0$) is a horizontal asymptote.
  • The domain is all real numbers, and the range is $y > 0$.

• The Exponential Function $y = e^x$

  • $e$ is Euler’s number, approximately equal to 2.71828.
  • $y = e^x$ is the natural exponential function.
  • It is its own derivative, i.e., $\frac{d}{dx}(e^x) = e^x$.

Diagram: A graph showing $y = 2^x$, $y = (\frac{1}{2})^x$, and $y = e^x$ illustrating exponential growth and decay.

Logarithmic Functions: $y = \log_a(x)$

Logarithmic functions are the inverse of exponential functions. The general form is $y = \log_a(x)$, where $a$ is the base of the logarithm, and $a > 0$ and $a \neq 1$.

• Key Features

  • The graph always passes through the point (1, 0) because $\log_a(1) = 0$.
  • If $a > 1$, the function is increasing.
  • If \$0 < a < 1$, the function is decreasing.
  • The y-axis ($x = 0$) is a vertical asymptote.
  • The domain is $x > 0$, and the range is all real numbers.

• Natural Logarithm: $y = \ln(x) = \log_e(x)$

  • The natural logarithm has base $e$.

• Common Logarithm: $y = \log_{10}(x)$

  • The common logarithm has base 10.

Diagram: A graph showing $y = \log_2(x)$, $y = \log_{\frac{1}{2}}(x)$, and $y = \ln(x)$ illustrating logarithmic growth and decay.

Circular Functions: $y = \sin(x)$, $y = \cos(x)$, $y = \tan(x)$

Circular functions (trigonometric functions) relate angles of a right triangle to ratios of its sides.

• Sine Function: $y = \sin(x)$

  • Domain: All real numbers.
  • Range: $[-1, 1]$.
  • Period: $2\pi$.
  • Amplitude: 1.
  • Odd function: $\sin(-x) = -\sin(x)$.

• Cosine Function: $y = \cos(x)$

  • Domain: All real numbers.
  • Range: $[-1, 1]$.
  • Period: $2\pi$.
  • Amplitude: 1.
  • Even function: $\cos(-x) = \cos(x)$.

• Tangent Function: $y = \tan(x) = \frac{\sin(x)}{\cos(x)}$

  • Domain: All real numbers except $x = \frac{(2n+1)\pi}{2}$, where $n$ is an integer (vertical asymptotes).
  • Range: All real numbers.
  • Period: $\pi$.
  • Vertical asymptotes at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.
  • Odd function: $\tan(-x) = -\tan(x)$.

Diagram: Graphs of $y = \sin(x)$, $y = \cos(x)$, and $y = \tan(x)$ showing their periodic nature and key features.

These notes provide a comprehensive overview of power, exponential, logarithmic, and circular functions, including their key features and graphs, which are essential for VCE Mathematical Methods.