Quadratic, Cubic, Reciprocal and Exponential Functions
Cubic Function
A polynomial function of degree three is called a cubic function. This function is of the form: f(x) = ax³ + bx² + cx + d; a,b,c,d ∈ R and a ≠ 0.
To sketch the graph of y = x³ (when a = 1), we need the shape, x and y intercepts and turning point. Here leading coefficient is 1 which is positive which shows that the graph will concave up in the first quadrant and concave down in the third quadrant. For each point (x,y) on the graph, the point (-x,-y) is also on the graph. At x = 0, y = 0, so this graph intersects both the axis at origin and is symmetric about origin. Also origin is the turning point of this graph. The domain and range of this function is the set of all real numbers.
Example: Sketch the graph of y = -2x³
Solution: y = -2x³. Here, the leading coefficient is -2 (negative), which shows that the graph will concave up in the second quadrant and concave down in the fourth quadrant. At x = 0, y = 0, so the graph passes through origin which is the turning point of the graph. For each point (-x,y) on the graph, the point (x,-y) is also on the graph, so this graph is symmetric about origin. The domain and range of this function is the set of all real numbers.
Square Root Function
The function defined by f(x) = √x, where x ≥ 0 is called a square root function. Its domain is the set of non-negative real numbers and its range is also the set of non-negative real numbers.