Introduction to Cubic Functions

• The parent function f(x) = x³ is shown.

• In factored form:
  f(x) = (x - 0)(x - 0)​(x - 0)

• Notice the repeated factor -> There's a repeated root

Graph of a Cubic Function

Multiple Choice

Blue is the parent function f(x) = x³.

Red is the translated function f(x) = (x - 2)³.

Which way did the graph shift?

• Up

• Down

• Left

• Right

• h moves the graph horizontally (left/right)

• f(x) = x³

• f(x) = (x - 2)³

• Think: "Opposite"

f(x) = a(x - h) + k

Poll

What do you think will happen to the graph when k is changed?

• It will "flip" over the y-axis

• It will "flip" over the x-axis

• It will shift up/down

• It will stretch along the y-axis

• k moves the graph vertically (up/down)

• f(x) = x³

• f(x) = (x)³ + 4

• Think: "Keep"

f(x) = a(x - h) + k

• f(x) = (x+2)³

• f(x) = (-x+2)³ flips horizontally across y-axis (y-intercept the same)

• f(x) = -(x+2)³ flips vertically along x-axis (x-intercept the same)

Reflections

• f(x) = x³ + x² + 1

• f(x) = (-x)³ + (-x)² + 1 flips horizontally across y-axis (y-intercept the same)

• f(x) = -(x³ + x² + 1) flips vertically along x-axis (x-intercept the same)

Reflections (Standard Form)

• f(x) = (x+2)³

• f(x) = (4x+2)³ shrinks by a factor of 1/4 horizontally along x-axis (y-intercept the same)

• f(x) = 4(x+2)³ stretches by a factor of 4vertically along y-axis (x-intercept the same)

Stretch/Shrink

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