Cubic graphs, sketching polynomials and asymptotes
Fill in and plot a cubic table, build a polynomial graph from its roots and its y-intercept, and draw the vertical and horizontal asymptotes of a rational function as lines.
- UK: Year 11
- Germany: grade 10
- Iran: grade 12
- Afghanistan: grade 10
- Syria: grade 11
- Palestine: grade 9
- Ukraine: grade 10
- Russia: grade 9
Guide
We build the graph of y = x3 − 3x from a table. Take care: the cube of a negative number stays negative, (−2)³ = −8. A cubic graph has an S shape and can cross the x-axis up to three times.
To sketch y = (x + 2)(x − 1)(x − 3) quickly: read the roots from the factors (x = −2, 1, 3; the sign flips) and find the y-intercept by putting x = 0.
In y = (2x + 1)/(x − 1) the denominator is zero at x = 1: the vertical line x = 1 is a vertical asymptote, which the curve hugs but never reaches. When x gets very large or very small the curve approaches the horizontal line y = 2 (the coefficients of x: 2 ÷ 1): a horizontal asymptote.