Cubics and transformations of graphs

Graphs of y = x³ and factorised cubics, x- and y-intercepts, and translations, reflections and dilations of any graph.

Cubics

$y = x^3$ rises from bottom left to top right with a flat point of inflection at the origin. A factorised cubic $y = a(x - p)(x - q)(x - r)$ crosses the $x$-axis at $p$, $q$ and $r$.

A factorised cubic

Positive a: starts low, ends high. Negative a flips it.

Example 1●●

Find the intercepts of $y = (x - 1)(x + 2)(x - 3)$.

$x$-intercepts: $1$, $-2$, $3$.

$y$-intercept: $(-1)(2)(-3) = 6$.

Transformations

Example 2●●

Describe $y = (x - 2)^3 + 1$ compared with $y = x^3$.

Shifted right $2$ and up $1$. Point of inflection $(2, 1)$.

Flashcards

$y = f(x) - 3$

Down $3$

$y = f(x + 4)$

Left $4$

$y = -f(x)$

Reflect in the $x$-axis

Practice

$y$-intercept of $y = (x - 1)(x + 2)(x - 3)$?
$x$-intercepts of $y = (x - 1)(x + 2)(x - 3)$?
Point of inflection of $y = (x - 2)^3 + 1$?
$y = f(x + 4)$ moves the graph of $y = f(x)$:
$y = -x^3$ is $y = x^3$:
$x$-intercepts of $y = x^2(x - 3)$?
$y$-intercept of $y = (x - 2)^3$?
Point of inflection of $y = (x + 1)^3 - 3$?