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Mathematical Methods

Functions and Graphs: Polynomial, Exponential, Logarithmic, and Trigonometric Functions

Year 11 · Year 12

  • ✓By the end of this lesson students will be able to identify and describe the key features of polynomial, exponential, logarithmic, and trigonometric functions.
  • ✓By the end of this lesson students will be able to sketch graphs of these functions, identifying intercepts, asymptotes, and turning points where applicable.
  • ✓By the end of this lesson students will be able to solve equations involving polynomial, exponential, logarithmic, and trigonometric functions algebraically and graphically.
  • ✓By the end of this lesson students will be able to apply knowledge of these functions to model and solve real-world problems.

Key concepts

Polynomial Functions

A polynomial function is a function of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_0 are real coefficients and n is a non-negative integer (the degree of the polynomial). Key features include the degree, leading coefficient (determines end behaviour), x-intercepts (roots), y-intercept, and turning points. The maximum number of x-intercepts is n, and the maximum number of turning points is n-1.

P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
Exponential Functions

An exponential function is a function of the form f(x) = a * b^x, where b > 0, b ≠ 1, and a ≠ 0. If b > 1, it represents exponential growth; if 0 < b < 1, it represents exponential decay. A key feature is the horizontal asymptote, typically y=0 for the basic form. The natural exponential function is f(x) = e^x, where e is Euler's number (approximately 2.718).

f(x) = a * b^x or f(x) = a * e^(kx)
Logarithmic Functions

A logarithmic function is the inverse of an exponential function. If y = b^x, then x = log_b(y). The function is written as f(x) = log_b(x), where b > 0 and b ≠ 1. Key features include a vertical asymptote (typically x=0 for the basic form), an x-intercept at (1,0), and a domain of x > 0. The natural logarithm is ln(x), which is log_e(x).

f(x) = log_b(x)
Trigonometric Functions

Trigonometric functions (sine, cosine, tangent) describe relationships between angles and sides of triangles, and are periodic. For y = A sin(B(x-C)) + D or y = A cos(B(x-C)) + D: |A| is the amplitude, 2π/|B| is the period, C is the phase shift, and D is the vertical shift. Tangent functions (y = A tan(B(x-C)) + D) have a period of π/|B| and vertical asymptotes.

y = A sin(B(x-C)) + D or y = A cos(B(x-C)) + D or y = A tan(B(x-C)) + D

Key facts to remember

  • 1The degree of a polynomial determines its maximum number of x-intercepts and turning points.
  • 2Exponential functions of the form y = a * b^x have a horizontal asymptote at y=0 (or y=D for y = a * b^x + D).
  • 3Logarithmic functions of the form y = log_b(x) have a vertical asymptote at x=0 (or x=C for y = log_b(x-C)).
  • 4The function y = log_b(x) is the inverse of y = b^x.
  • 5Trigonometric functions (sine, cosine, tangent) are periodic, meaning their graphs repeat over regular intervals.
  • 6In Mathematical Methods, angles for trigonometric functions are typically measured in radians.
  • 7Euler's number, e ≈ 2.718, is the base for the natural exponential and logarithmic functions.
  • 8The domain of log_b(x) requires x > 0.

Worked examples

Example 1

Sketch the graph of the polynomial function y = (x-1)^2 (x+2), identifying all intercepts and describing its end behaviour.

IFind x-intercepts: Set y = 0. (x-1)^2 (x+2) = 0. This gives x = 1 (multiplicity 2) and x = -2 (multiplicity 1).
IIFind y-intercept: Set x = 0. y = (0-1)^2 (0+2) = (-1)^2 (2) = 1 * 2 = 2. So, the y-intercept is (0, 2).
IIIDetermine end behaviour: The highest power term is x^2 * x = x^3. Since the leading coefficient is positive (1) and the degree is odd (3), as x -> ∞, y -> ∞, and as x -> -∞, y -> -∞.
IVConsider behaviour at x-intercepts: At x = 1 (multiplicity 2), the graph touches the x-axis and turns around. At x = -2 (multiplicity 1), the graph crosses the x-axis.
VSketch the graph using these points and behaviours. The graph crosses at (-2,0), passes through (0,2), touches at (1,0), and follows the cubic end behaviour.

Answer

The graph has x-intercepts at (-2, 0) and (1, 0) (touching point), and a y-intercept at (0, 2). As x -> ∞, y -> ∞; as x -> -∞, y -> -∞. The graph crosses the x-axis at x=-2 and touches the x-axis at x=1.

Finding the exact coordinates of turning points for cubic functions typically requires calculus (finding where the derivative is zero).

Example 2

Solve the following equations: a) 3^(2x-1) = 27 b) log_2(x+3) = 4

Ia) For 3^(2x-1) = 27:
II Rewrite 27 as a power of 3: 27 = 3^3.
III So, 3^(2x-1) = 3^3.
IV Equate the exponents: 2x-1 = 3.
V Add 1 to both sides: 2x = 4.
VI Divide by 2: x = 2.
VIIb) For log_2(x+3) = 4:
VIII Rewrite the logarithmic equation in exponential form: x+3 = 2^4.
9 Calculate 2^4: 2^4 = 16.
10 So, x+3 = 16.
11 Subtract 3 from both sides: x = 13.
12 Check domain: For log_2(x+3), we need x+3 > 0, so x > -3. Our solution x=13 satisfies this condition.

Answer

a) x = 2 b) x = 13

Always check the domain for logarithmic equations to ensure the solution is valid.

Example 3

For the trigonometric function y = 3 sin(2x - π) + 1, determine the amplitude, period, phase shift, and vertical shift.

IThe general form for a sine function is y = A sin(B(x-C)) + D.
IIRewrite the given equation to match this form: y = 3 sin(2(x - π/2)) + 1.
IIIIdentify A: A = 3. The amplitude is |A| = |3| = 3.
IVIdentify B: B = 2. The period is 2π/|B| = 2π/2 = π.
VIdentify C: C = π/2. The phase shift is C = π/2 to the right.
VIIdentify D: D = 1. The vertical shift is D = 1 unit upwards.

Answer

Amplitude = 3, Period = π, Phase shift = π/2 to the right, Vertical shift = 1 unit upwards.

Ensure to factor out B from the argument of the trigonometric function to correctly identify the phase shift C.

Common mistakes

  • ✗Incorrectly identifying asymptotes for exponential and logarithmic functions, especially after transformations.
  • ✗Making errors when applying logarithm laws or exponent laws during equation solving.
  • ✗Forgetting the domain restriction for logarithmic functions (argument must be positive) when solving equations.
  • ✗Incorrectly determining the phase shift of trigonometric functions by not factoring out the 'B' value (e.g., in sin(Bx+C), the phase shift is -C/B, not -C).
  • ✗Not considering the multiplicity of roots when sketching polynomial graphs, leading to incorrect behaviour at x-intercepts (crossing vs. touching).

Exam tips

  • ★Always check the domain and range of functions, particularly for logarithmic functions, as solutions outside the domain are invalid.
  • ★Use graphing calculators to verify your sketches and solutions, but ensure you show all algebraic working steps as required for marks.
  • ★Memorise key exact values for trigonometric functions (e.g., sin(π/6), cos(π/4), tan(π/3)) and their corresponding angles in radians.
  • ★Practice solving equations that combine different types of functions (e.g., exponential and linear, or trigonometric and quadratic).
  • ★Pay close attention to units, especially when working with trigonometric functions (radians are standard in calculus-based maths).

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