1. Stretches
The graph of y + 3 = f(x) is the graph of f(x) translated…
 up 3 units  left 3 units  down 3 units  right 3 units x
2. The graph of f(x) + 4 is the graph of f(x) translated…
 4 units up  4 units left  4 units down  4 units right x
3. The graph of f(x – 7) + 6 is the graph of f(x) that has been translated..
 7 units left, 6 units up  7 units left, 6 units down
 7 units right, 6 units down  7 units right, 6 units up x
4. In general, the graph of f(x – h) + k, where h and k are positive, as compared to the parent function graph f(x), is translated
 h units left and k units up  h units right and k units up
 h units left and k units down  h units right and k units down x
Math 30-1

2. Investigating Vertical Stretches: Recall the affect of “a” for a Quadratic
Math 30-1

3. Vertical Stretches: The effect of parameter a.
A stretch changes the shape of the graph.
(Translations changed the position of a graph.)
Function Transformations: Vertical Stretch
|a| describes a vertical stretch about the x-axis.
An Invariant point is a point on a graph that remains unchanged after a transformation.
1.2A Vertical Stretches
Math 30-1

4. Vertical Stretches
In general, for any function y = f(x), the graph of the function
y =a f(x) has been vertically stretched about the x-axis
by a factor of |a| .
The point (x, y) → (x, ay). Only the y coordinates are affected.
Invariant points are on the line of stretch, the x-axis.
are the x-intercepts.
When |a| > 1, the points on the graph move farther away from the x-axis.
Vertical stretch by a factor of 3
When |a|< 1, the points on the graph move closer to the x-axis.
Vertical stretch by a factor of ⅓
Math 30-1

5. Vertical Stretching y = af(x), |a| > 1
y = 2f(x)
A vertical stretch
about the x-axis by
a factor of 2.
y = f(x)
Key Points
(x, y) →
(x, 2y)
(-2, 0) →
(-2, 0)
(-1, -7) →
(-1, -14)
(1/2, 1) →
(1/2, 2)
Invariant Points
Domain and Range
(-2, 0)
(0, 0)
(1, 0)
(2, 0)
Math 30-1

6. Vertical Stretches about the x-axis y = af(x), |a| < 1
Consider
Write the equation of the function after a vertical stretch about the x-axis by a factor of ½.
Write the transformation in function notation.
Write the coordinates of the image of the point (-7, 7)
(-7, 3.5)
The point (x, y) maps to
List any invariant points.
How are the domain and range affected?
Math 30-1

7. The graph of g(x) is a transformation of f(x).
Horizontal stretch
By a factor of ½
(2, 0)→
(1, 0)
(x, y)→(½x, y)
Invariant Point
On the y-axis
Is the transformation a translation?
Is the transformation a vertical stretch?
Math 30-1

8. Horizontal Stretches
In general, for any function y = f(x), the graph of the function
y = f(bx) has been horizontally stretched by a factor of
The point (x, y) →
Only the x coordinates are affected.
Invariant points are on the line of stretch, the y-axis.
are the y-intercepts.
When |b| > 1, the points on the graph move closer to the y-axis.
Horizontal stretch by a factor of ⅓
When |b|< 1, the points on the graph move farther away from the x-axis.
Horizontal stretch by a factor of 4
Math 30-1

9. Characteristics of Horizontal Stretches about the y-axis
Consider
Write the equation of the function after a horizontal stretch about the y-axis by a factor of 3.
Write the transformation in function notation.
Can it be written in any other way?
Write the coordinates of the image of the point (-3, 3 )
→ (-9, 3)
Write the coordinates of the image of the point (x, y)
List any invariant points.
How are the domain and range affected?
Math 30-1

10. Is This a Horizontal or Vertical Stretch of y = f(x)?
The graph y = f(x) is stretched vertically about the x-axis by
a factor of
Math 30-1

11. Has the graph been stretched Horizontally or Vertically?
(1, 1)
Math 30-1

12. Zeros of a Function 1. What are the zeros of the function?
x = -2, 0, 3
2. Use transformations to determine the zeros of the following functions.
x = -1, 1, 4
x = -2, 0, 3
x = -4, 0, 6
Math 30-1

13. Describing the Horizontal or Vertical Stretch of a Function
Describe the transformation in words compared to the graph of a function y = f(x).
a) y = f(3x)
b) 3y = f(x)
Stretched horizontally by a
factor of about the y-axis.
Stretch vertically by a factor
of about the x-axis.
c) y = f( x)
d) y = 2f(x)
Stretched vertically by a factor
of about the x-axis.
Stretched horizontally by a factor
of 2 about the y-axis.
Math 30-1

14. Consider the graph of a function y = f(x)
The transformation described by y = f(2x+4) is horizontal stretch about the y-axis by a factor of ½.
The translation described by y = f(2x + 4) is horizontal shift of 4 units to the left.
y = f(2(x + 2))
Translations must be factored
Math 30-1

15. Stating the Equation of y = af(kx)
The graph of the function y = f(x) is transformed as described.
Write the new equation in the form y = af(bx).
Stretched horizontal by a factor of one-third about the y-axis, and stretched vertically about the x-axis by a factor of two.
y = 2f(3x)
Stretched horizontally by a factor of two about the y-axis and translated four units to the left.
Math 30-1

16. Assignment
Page 28
2, 5a,b, 6, 7a,c, 8, 13, 14c, d
Math 30-1

17. Write the equation of the function after a horizontal stretch about the y-axis by a factor of 3.
Check
Verify by substitution
Math 30-1