Chapter 1: Quadratic Functions and Equations in One Variable
Form 4 Mathematics Bab 1: Quadratic Functions and Equations in One Variable
1.1 Quadratic Expressions
A quadratic expression in one variable is an expression whose highest power of the variable is two. (p.3)
General form: ax2 + bx + c, where a, b and c are constants, a ≠ 0, and x is a variable. (p.3)
The values of the constants b and c can be zero; letters other than x can be used as the variable. (p.3)
Not a quadratic expression: x3 − 6 (highest power is 3) and 3x2 + 2y + 1 (two variables). (p.3)
Not a quadratic expression: a variable with a power that is not a whole number, e.g. 3/x2 (= 3x−2) or x1/2 (= √x). (p.3)
In ax2 + bx + c: a is the coefficient of x2, b is the coefficient of x, c is a constant. (p.4)
1.1 Quadratic Functions — Shape and Characteristics
A quadratic expression is written ax2 + bx + c; a quadratic function is written f(x) = ax2 + bx + c. (p.5)
A quadratic function is a many-to-one relation: two different objects have the same image, except at the maximum or minimum point. (p.6)
The curved shape of the graph of a quadratic function is a parabola. There are only two shapes, decided by the value of a. (p.7)
a > 0: graph has a ∪ shape (opens upward). a < 0: graph has a ∩ shape (opens downward). (p.7)
For a < 0 the graph has a maximum point (highest y-coordinate); for a > 0 it has a minimum point (lowest y-coordinate). (p.7)
The maximum or minimum point is also called a stationary point or turning point. (p.8)
Axis of symmetry: a straight line parallel to the y-axis that divides the graph into two parts of the same size and shape; it passes through the maximum or minimum point. (p.8)
The equation of the axis of symmetry is x = −b / 2a. (p.8)
The equation of a straight line parallel to the y-axis is x = h (h is a constant). Each graph has one axis of symmetry, passing through its maximum/minimum point (h, k) or (m, n). (p.9)
1.1 Effects of a, b and c on f(x) = ax2 + bx + c
The value of a determines the shape of the graph and affects its width. (p.11)
The smaller the value of a, the wider the curved shape of the graph, and vice versa. (p.11)
The value of b determines the position of the axis of symmetry. (p.12)
If a > 0: b > 0 → axis of symmetry lies left of the y-axis; b < 0 → right of the y-axis; b = 0 → the y-axis. (p.12)
If a < 0: b > 0 → axis of symmetry lies right of the y-axis; b < 0 → left of the y-axis; b = 0 → the y-axis. (p.12)
The value of c is the y-intercept of the graph (the point where the graph meets the y-axis). (p.13)
If the graph passes through (0, k), then c = k. Otherwise substitute the given point into the function to find c. (p.13)
1.1 Quadratic Equations
A quadratic function is f(x) = ax2 + bx + c; a quadratic equation is written in the general form ax2 + bx + c = 0. (p.15)
To form a quadratic equation from a situation, write an expression (e.g. area) in terms of the variable, then equate it to the given value and rearrange to the general form. (p.15)
The roots of a quadratic equation ax2 + bx + c = 0 are the values of the variable that satisfy the equation. (p.16)
There are one or two values of the variable that satisfy a quadratic equation. (p.17)
The roots are the points of intersection of the graph of f(x) = ax2 + bx + c with the x-axis (the x-intercepts). (p.18)
To test whether a value is a root, substitute it into the equation: it is a root if left side = right side. (p.19)
1.1 Solving Quadratic Equations
Factorisation method: first write the equation in the form ax2 + bx + c = 0, factorise, then equate each factor to zero. (p.21)
Example: x2 − 5x + 6 = 0 → (x − 3)(x − 2) = 0 → x = 3 or x = 2. (p.21)
Equations with fractions: multiply out to remove the denominator, then rearrange to general form before factorising. (p.21)
A quadratic equation can also be solved by completing the square or by the formula x = (−b ± √(b2 − 4ac)) / 2a. (p.21)
Graphical method: read the values of x at the points where the graph of f(x) = ax2 + bx + c intersects the x-axis. (p.22)
1.1 Sketching Graphs of Quadratic Functions
A sketch must show: (1) the correct shape, (2) the y-intercept, (3) the x-intercepts or one point that the graph passes through. (p.23)
Case 1 (graph intersects the x-axis): use a for the shape, c for the y-intercept, and solve f(x) = 0 for the x-intercepts. (p.24)
Case 2 (graph does not intersect the x-axis): if b = 0 the y-axis is the axis of symmetry and the turning point is (0, c); plot one extra point (e.g. x = 2). (p.25)
If b = 0 for a quadratic function, then the y-axis is the axis of symmetry of its graph. (p.25)
1.1 Problem Solving
Strategy: write the expression (e.g. surface area) in terms of x, form the quadratic equation, solve it, then determine the measurements and actual value. (p.26)
Reject any root that gives a negative length, since a measurement of length cannot be negative. (p.26)