PRE-CALCULUS UNIT 2 NOTES
TRANSFORMATIONS OF GRAPHS
Start with: y = f(x)
Vertical Shift Up:
y = f(x) + k
Move graph UP k units.
Example:
y = x² + 3
Up 3
Vertical Shift Down:
y = f(x) - k
Move graph DOWN k units.
Example:
y = x² - 4
Down 4
Horizontal Shift Right:
y = f(x - h)
Move graph RIGHT h units.
Example:
y = (x - 2)²
Right 2
Horizontal Shift Left:
y = f(x + h)
Move graph LEFT h units.
Example:
y = (x + 5)²
Left 5
Reflection Across x-axis:
y = -f(x)
Example:
y = -x²
, Reflection Across y-axis:
y = f(-x)
Example:
y = (-x)²
Vertical Stretch:
y = af(x)
a>1
Example:
y = 3x²
Vertical Compression:
y = af(x)
0<a<1
Example:
y = (1/2)x²
Example of Multiple Transformations:
y = -2(x - 3)² + 4
Right 3
Vertical Stretch by 2
Reflect over x-axis
Up 4
ANALYZING GRAPHS
x-intercept:
Point where graph crosses x-axis.
Set y = 0.
y-intercept:
Point where graph crosses y-axis.
Set x = 0.
Increasing:
Graph rises from left to right.
Decreasing:
TRANSFORMATIONS OF GRAPHS
Start with: y = f(x)
Vertical Shift Up:
y = f(x) + k
Move graph UP k units.
Example:
y = x² + 3
Up 3
Vertical Shift Down:
y = f(x) - k
Move graph DOWN k units.
Example:
y = x² - 4
Down 4
Horizontal Shift Right:
y = f(x - h)
Move graph RIGHT h units.
Example:
y = (x - 2)²
Right 2
Horizontal Shift Left:
y = f(x + h)
Move graph LEFT h units.
Example:
y = (x + 5)²
Left 5
Reflection Across x-axis:
y = -f(x)
Example:
y = -x²
, Reflection Across y-axis:
y = f(-x)
Example:
y = (-x)²
Vertical Stretch:
y = af(x)
a>1
Example:
y = 3x²
Vertical Compression:
y = af(x)
0<a<1
Example:
y = (1/2)x²
Example of Multiple Transformations:
y = -2(x - 3)² + 4
Right 3
Vertical Stretch by 2
Reflect over x-axis
Up 4
ANALYZING GRAPHS
x-intercept:
Point where graph crosses x-axis.
Set y = 0.
y-intercept:
Point where graph crosses y-axis.
Set x = 0.
Increasing:
Graph rises from left to right.
Decreasing:
