Algebra & Functions

Functions: Linear, Quadratic, Exponential, Hyperbolic, and Applications

Grade 10 · Grade 11 · Grade 12

✓By the end of this lesson students will be able to identify and describe the properties of linear, quadratic, exponential, and hyperbolic functions.
✓By the end of this lesson students will be able to sketch graphs of these functions, indicating intercepts, turning points, and asymptotes.
✓By the end of this lesson students will be able to determine the equations of these functions from given information or graphs.
✓By the end of this lesson students will be able to apply function concepts to solve real-world problems involving financial mathematics (growth and decay).

Key concepts

Linear Function

A linear function produces a straight-line graph. It has a constant rate of change (gradient).

f(x) = mx + c or y = mx + c\nWhere 'm' is the gradient and 'c' is the y-intercept.

Quadratic Function

A quadratic function produces a parabolic graph (U-shaped or inverted U-shaped). It has a single turning point (vertex) and an axis of symmetry.

f(x) = ax^2 + bx + c or y = ax^2 + bx + c\nWhere 'a', 'b', 'c' are constants and a ≠ 0.\nIf a > 0, the parabola opens upwards. If a < 0, it opens downwards.\nThe x-coordinate of the turning point is given by x = -b/(2a).

Exponential Function

An exponential function shows rapid growth or decay. Its graph approaches a horizontal asymptote but never touches it.

f(x) = ab^x + q or y = ab^x + q\nWhere 'a' and 'b' are constants, b > 0 and b ≠ 1.\n'q' is the horizontal asymptote, y = q.

Hyperbolic Function

A hyperbolic function consists of two separate branches. Its graph has both a vertical and a horizontal asymptote.

f(x) = a/(x-p) + q or y = a/(x-p) + q\nWhere 'a', 'p', 'q' are constants and a ≠ 0.\n'p' is the vertical asymptote, x = p.\n'q' is the horizontal asymptote, y = q.

Finance Applications (Growth & Decay)

These are real-world applications of functions, particularly exponential functions, to calculate interest, depreciation, population growth, etc.

Simple Interest: A = P(1 + in)\nCompound Interest: A = P(1 + i)^n\nStraight Line Depreciation: A = P(1 - in)\nReducing Balance Depreciation: A = P(1 - i)^n\nWhere A = accumulated amount, P = principal amount, i = interest rate per period, n = number of periods.

Key facts to remember

1A function assigns each input (domain value) to exactly one output (range value).
2Linear functions have a constant gradient (m) and a straight-line graph.
3Quadratic functions form parabolas with a single turning point at x = -b/(2a).
4Exponential functions have a horizontal asymptote at y = q.
5Hyperbolic functions have both a vertical asymptote (x = p) and a horizontal asymptote (y = q).
6Asymptotes are lines that the graph approaches but never touches.
7Compound interest generally yields higher returns than simple interest over longer periods due to interest being earned on accumulated interest.
8The domain of a function refers to all possible input (x) values, and the range refers to all possible output (y) values.

Worked examples

Example 1

Sketch the graph of f(x) = x^2 - 4x - 5, clearly indicating all intercepts with the axes and the turning point.

I1. Determine the y-intercept (set x = 0):\n f(0) = (0)^2 - 4(0) - 5 = -5\n So, the y-intercept is (0; -5).

II2. Determine the x-intercepts (set f(x) = 0):\n x^2 - 4x - 5 = 0\n (x - 5)(x + 1) = 0\n x = 5 or x = -1\n So, the x-intercepts are (5; 0) and (-1; 0).

III3. Determine the x-coordinate of the turning point:\n x = -b/(2a) = -(-4)/(2*1) = 4/2 = 2

IV4. Determine the y-coordinate of the turning point (sub x = 2 into f(x)):\n f(2) = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9\n So, the turning point is (2; -9).

V5. Plot the intercepts and turning point, then draw a smooth parabola through them.

Answer

The graph is a parabola opening upwards with y-intercept (0; -5), x-intercepts (5; 0) and (-1; 0), and turning point (2; -9).

Always label all key points on your sketch.

Example 2

Determine the equation of the hyperbola g(x) = a/(x-p) + q if its asymptotes are x = 2 and y = -1, and it passes through the point (3; 1).

I1. Use the asymptotes to find p and q:\n The vertical asymptote is x = p, so p = 2.\n The horizontal asymptote is y = q, so q = -1.\n Substitute these into the general equation: g(x) = a/(x-2) - 1.

II2. Substitute the given point (3; 1) into the equation to find a:\n 1 = a/(3-2) - 1\n 1 = a/1 - 1\n 1 = a - 1\n a = 2

III3. Write the final equation:\n g(x) = 2/(x-2) - 1

Answer

g(x) = 2/(x-2) - 1

Remember that 'p' affects the vertical asymptote and 'q' affects the horizontal asymptote.

Example 3

R15 000 is invested at an interest rate of 8% per annum compounded quarterly. Calculate the value of the investment after 5 years.

I1. Identify the given values:\n Principal amount (P) = R15 000\n Nominal interest rate = 8% p.a. = 0.08\n Compounding period = quarterly (4 times a year)\n Time (years) = 5

II2. Adjust the interest rate (i) and number of periods (n) for quarterly compounding:\n i = (0.08) / 4 = 0.02 per quarter\n n = 5 years * 4 quarters/year = 20 quarters

III3. Apply the compound interest formula: A = P(1 + i)^n\n A = 15000(1 + 0.02)^20

IV4. Calculate the final amount:\n A = 15000(1.02)^20\n A ≈ 15000 * 1.485947396\n A ≈ 22289.21

Answer

The value of the investment after 5 years will be R22 289.21.

Always ensure the interest rate 'i' and the number of periods 'n' match the compounding frequency.

Common mistakes

✗Confusing the roles of 'p' and 'q' in hyperbolic and exponential functions, especially regarding which asymptote is vertical or horizontal.
✗Incorrectly adjusting the interest rate (i) and number of periods (n) for non-annual compounding in financial calculations.
✗Failing to indicate all necessary intercepts, turning points, and asymptotes when sketching graphs.
✗Assuming the horizontal asymptote for an exponential function is always y=0, instead of y=q.
✗Making algebraic errors when solving for intercepts or the turning point coordinates.

Exam tips

★Always show all your working steps clearly, as marks are often awarded for method.
★When sketching graphs, ensure all intercepts, turning points, and asymptotes are accurately calculated and clearly labelled.
★Read financial mathematics questions carefully to identify the correct formula (simple/compound interest, straight-line/reducing balance depreciation) and the compounding period.
★Use your calculator efficiently for calculations, but write down the formula and the substitution before the final answer.
★Practice identifying the type of function from its equation and understanding how parameters (a, b, c, m, p, q) affect the graph.

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