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Graphs of polynomials and rational functions
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A quick sketch shows the shape, the intercepts and any asymptotes. You do not need to plot dozens of points.
Cubic graphs
y = (x − 1)(x + 2)(x − 4) crosses the x-axis at x = 1, −2 and 4.
Its y-intercept is (−1)(2)(−4) = 8.
Positive x3 coefficient: starts bottom left, ends top right.
Negative x3 coefficient: starts top left, ends bottom right.
A squared bracket, like (x − 2)2, means the graph touches the axis there.
Its y-intercept is (−1)(2)(−4) = 8.
Positive x3 coefficient: starts bottom left, ends top right.
Negative x3 coefficient: starts top left, ends bottom right.
A squared bracket, like (x − 2)2, means the graph touches the axis there.
Rational functions
y = (ax + b)(cx + d)
Vertical asymptote: where the bottom is zero, x = −dc
Horizontal asymptote: y = ac (for very large x only the x terms matter)
y = 3(x − 2) has asymptotes x = 2 and y = 0.
Vertical asymptote: where the bottom is zero, x = −dc
Horizontal asymptote: y = ac (for very large x only the x terms matter)
y = 3(x − 2) has asymptotes x = 2 and y = 0.
Intercepts
y-intercept: put x = 0.
x-intercept: put y = 0, which means the top is zero.
y = (2x − 6)(x + 1): crosses the x-axis at x = 3 and the y-axis at y = −6.
x-intercept: put y = 0, which means the top is zero.
y = (2x − 6)(x + 1): crosses the x-axis at x = 3 and the y-axis at y = −6.
Find the asymptotes and intercepts of y = (x + 4)(x − 2).
- Bottom zero: x = 2
- Large x: y → 11 = 1
- x = 0 gives y = −2; y = 0 gives x = −4
Answer: Asymptotes x = 2 and y = 1; intercepts (0, −2) and (−4, 0)
Polynomials: find the roots, the y-intercept and the end behaviour. y = (ax + b)(cx + d): asymptotes x = −dc and y = ac.
Check you have got it
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