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.