Section 3.5 Transformations of functions
1) Let f(x) = |𝑥|
1a) Sketch a graph of f(x)
x
f(x)
point
-2
𝑓(−2) = |−2| = 2
(-2,2)
-1
𝑓(−1) = |−1| = 1
(-1,1)
0
𝑓(0) = |0| = 0
(0,0)
1
𝑓(1) = |1| = 1
(1,1)
2
𝑓(2) = |2| = 2
(2,2)
1b) Find f(x-3) = |𝑥 − 3|
1c) Make a table of values and sketch a graph of f(x-3) on the same graph as the graph of f(x)
x
Graph of f(x-3) = |𝑥 − 3| shown in blue
f(x-3)
1
𝑓(1) = |1 − 3| = 2
(1,2)
2
𝑓(2) = |2 − 3| = 1
(2,1)
3
𝑓(3) = |3 − 3| = 0
(3,0)
4
𝑓(4) = |4 − 3| = 1
(4,1)
5
𝑓(5) = |5 − 3| = 2
(5,2)
1d) Describe the transformation to obtain the graph of f(x-3) from the graph of f(x)
Same shape, but shifted 3 units to the right.
point
1e) Find f(x+2) = |𝑥 + 2|
1f) Make a table of values and sketch a graph of f(x+2) on the same graph you have already created
Graph of f(x+2) = |𝑥 + 2| shown in blue
x
f(x+2)
point
-4
𝑓(−4) = |−4 + 2| = 2
(-4,2)
-3
𝑓(−3) = |−3 + 2| = 1
(-3,1)
-2
𝑓(−2) = |−2 + 2| = 0
(-2,0)
-1
𝑓(−1) = |−1 + 2| = 1
(-1,1)
0
𝑓(0) = |0 + 2| = 2
(0,2)
1g) Describe the transformation to obtain the graph of f(x+2) from the graph of f(x)
Shifted two units to the left.
3) Let f(x) = 𝑥 2
3a) Sketch a graph of f(x)
x
f(x)
point
-2
f(-2) = (-2)2 = 4
(-2, 4)
-1
f(-1) = (-1)2 = 1
(-1, 1)
0
f(0) = (0)2 = 0
(0, 0)
1
f(1) = (1)2 = 1
(1, 1)
2
f(2) = (2)2 = 4
(2, 4)
x
f(x - 2)
point
0
f(0) = (0-2)2 = 4
(0, 4)
1
f(1) = (1 – 2)2 = 1
(1, 1)
2
f(2) = (2 – 2)2 = 0
(2, 0)
3
f(3) = (3 – 2)2 = 1
(3, 1)
4
f(4) = (4 - 2)2 = 4
(4, 4)
3b) Find f(x-2) = (x – 2)2
3c) Make a table of values and sketch a graph of f(x-2)
Graph of f(x-2) = (x – 2)2 shown in blue
3d) Describe the transformation to obtain the graph of f(x-2) from the graph of f(x)
Same shape as f(x) but shifted 2 units to the right.
3e) Find f(x+1) = (x + 1)2
3f) Make a table of values and sketch a graph of f(x+1) on the same graph you have already created
x
f(x + 1)
point
-3
f(-3) = (-3 + 1)2 = 4
(-3, 4)
-2
f(-2) = (-2 + 1)2 = 1
(-2, 1)
-1
f(-1) = (-1 + 1)2 = 0
(-1, 0)
0
f(0) = (0 + 1)2 = 1
(0, 1)
1
f(1) = (1 + 1)2 = 4
(1, 4)
Graph of f(x +1) = (x+1)2 shown in blue
3g) Describe the transformation to obtain the graph of f(x+1) from the graph of f(x)
Same shape as f(x) = x2 but shifted one unit to the left
5) Let f(x) = √𝑥 + 2
5a) Sketch a graph of f(x)
x
f(x)
point
0
𝑓(0) = √0 = 0
(0, 0)
1
𝑓(1) = √1 = 1
(1, 1)
2
𝑓(2) = √2 = 1.41
(2, 1.41)
3
𝑓(3) = √3 = 1.73
(3, 1.73)
4
𝑓(4) = √4 = 2
(4, 2)
5b) Find f(x-3) = √𝑥 − 3
5c) Make a table of values and sketch a graph of f(x-3)
x
f(x-3)
3
𝑓(3) = √3 − 3 = 0
(3, 0)
4
𝑓(4) = √4 − 3 = 1
(4, 1)
5
𝑓(5) = √5 − 3 = 1.41
(5, 1.41)
6
𝑓(6) = √6 − 3 = 1.73
(6, 1.73)
7
𝑓(7) = √7 − 3 = 2
Graph of f(x-3) shown in blue
5d) Describe the transformation to obtain the graph of f(x-3) from the graph of f(x)
Same shape as f(x) but shifted 3 units to the right.
point
(7, 2)
5e) Find f(x+1) = √𝑥 + 1
5f) Make a table of values and sketch a graph of f(x+1) on the same graph you have already created
graph of f(x + 1) shown in blue
x
f(x+1)
point
-1
𝑓(−1) = √−1 + 1 = 0
(-1, 0)
0
𝑓(0) = √0 + 1 = 1
(0, 1)
1
𝑓(1) = √1 + 1 = 1.41
(1, 1.41)
2
𝑓(2) = √2 + 1 = 1.73
(2, 1.73)
3
𝑓(3) = √3 + 1 = 2
5g) Describe the transformation to obtain the graph of f(x+1) from the graph of f(x)
Same shape as f(x) but shifted 1 unit to the left.
(3, 2)
7) Let f(x) = |𝑥 + 3|
7a) Sketch a graph of f(x)
x
f(x)
point
-5
f(-5) = |−5 + 3|= 2
(-5, 2)
-4
f(-4) = |−4 + 3|= 1
(-4, 1)
-3
f(-3) = |−3 + 3|= 0
(-3, 0)
-2
f(-2) = |−2 + 3| = 1
(-2, 1)
-1
f(-1) = |−1 + 3|= 2
(-1, 2)
7b) Find f(x) – 3 = |𝑥 + 3| − 3
7c) Make a table of values and sketch a graph of f(x) – 3 on the same graph as the graph of f(x)
Graph of f(x) – 3 shown in blue
x
f(x) – 3
point
-5
f(-5) = |−5 + 3| − 3= -1
(-5, -1)
-4
f(-4) = |−4 + 3| − 3= -2
(-4, -2)
-3
f(-3) = |−3 + 3| − 3= -3
(-3, -3)
-2
f(-2) = |−2 + 3| − 3 = -2
(-2, -2)
-1
f(-1) = |−1 + 3| − 3= -1
(-1, -1)
7d) Describe the transformation to obtain the graph of f(x) – 3 from the graph of f(x)
Same shape as f(x) but shifted 3 units to the down.
7e) Find f(x) + 2 = |𝑥 + 3| + 2
7f) Make a table of values and sketch a graph of f(x) + 2 on the same graph you have already created
Graph of f(x) + 2 = |𝑥 + 3| + 2 shown in blue
x
f(x) + 2
point
-5
f(-5) = |−5 + 3| + 2= 4
(-5, 4)
-4
f(-4) = |−4 + 3| + 2= 3
(-4, 3)
-3
f(-3) = |−3 + 3| + 2= 2
(-3, 2)
-2
f(-2) = |−2 + 3| + 2 = 3
(-2, 3)
-1
f(-1) = |−1 + 3| + 2= 4
(-1, 4)
7g) Describe the transformation to obtain the graph of f(x+2) from the graph of f(x)
Same shape as f(x) but shifte 2 units up.
9) Let f(x) = (𝑥 − 3)2
9a) Sketch a graph of f(x)
x
f(x)
point
1
f(1) = (1 – 3)2 = 4
(1, 4)
2
f(2) = (2 – 3)2 = 1
(2, 1)
3
f(3) = (3 – 3)2 = 0
(3, 0)
4
f(4) = (4 – 3)2 = 1
(4, 1)
5
f(5) = (5 – 3)2 = 4
(5, 4)
x
f(x) - 2
point
1
f(1) = (1 – 3)2 – 2 = 2
(1, 2)
2
f(2) = (2 – 3)2 – 2 = -1
(2, -1)
3
f(3) = (3 – 3)2 – 2 = -2
(3, -2)
4
f(4) = (4 – 3)2 – 2 = -1
(4, -1)
5
f(5) = (5 – 3)2 – 2 = 2
(5, 2)
9b) Find f(x) – 2 = (x – 3)2 - 2
9c) Make a table of values and sketch a graph of f(x) – 2
Graph of f(x) - 2 shown in blue
9d) Describe the transformation to obtain the graph of f(x) – 2 from the graph of f(x)
Same shape as f(x) but shifted down 2 units.
9e) Find f(x) + 1 = (x – 3)2 + 1
9f) Make a table of values and sketch a graph of f(x) + 1 on the same graph you have already created
x
f(x) +1
point
1
f(1) = (1 – 3)2 +1 = 5
(1, 5)
2
f(2) = (2 – 3)2 + 1 = 2
(2, 2)
3
f(3) = (3 – 3)2 + 1 = 1
(3, 1)
4
f(4) = (4 – 3)2 +1 = 2
(4, 2)
5
f(5) = (5 – 3)2 + 1 = 5
(5, 5)
Graph of f(x) + 1 shown in blue
9g) Describe the transformation to obtain the graph of f(x) + 1 from the graph of f(x)
Same shape as f(x) but shifted up 1 unit.
11) Let f(x) = √𝑥 − 4
11a) Sketch a graph of f(x)
x
f(x)
Point
4
𝑓(4) = √4 − 4 = 0
(4, 0)
5
𝑓(5) = √5 − 4 = 1
(5, 1)
6
𝑓(6) = √6 − 4 = 1.41
(6, 1.41)
7
𝑓(7) = √7 − 4 = 1.73
(7, 1.73)
8
𝑓(8) = √8 − 4 = 2
(8, 2)
11b) Find f(x) – 3 = √𝑥 − 4 − 3
11c) Make a table of values and sketch a graph of f(x) – 3
x
Graph of f(x) - 3 shown in blue
f(x) – 3
4
𝑓(4) = √4 − 4 − 2 = −3
(4, -3)
5
𝑓(5) = √5 − 4 − 3 = −2
(5, -2)
6
𝑓(6) = √6 − 4 − 3 = −1.59
(6, -1.59)
7
𝑓(7) = √7 − 4 − 3 = −1.27
(7, -1.27)
8
𝑓(8) = √8 − 4 − 3 = −1
11d) Describe the transformation to obtain the graph of f(x) – 3 from the graph of f(x)
Same shape as f(x) but shifted down 3 units
Point
(8, -1)
11e) Find f(x) + 1 = √𝑥 − 4 + 1
11f) Make a table of values and sketch a graph of f(x) + 1 on the same graph you have already created
x
Graph of f(x) + 1 shown in blue
f(x) + 1
4
𝑓(4) = √4 − 4 + 1 = 1
(4, 1)
5
𝑓(5) = √5 − 4 + 1 = 2
(5, 2)
6
𝑓(6) = √6 − 4 + 1 = 2.41
(6, 2.41)
7
𝑓(7) = √7 − 4 + 1 = 2.73
(7, 2.73)
8
𝑓(8) = √8 − 4 + 1 = 3
11g) Describe the transformation to obtain the graph of f(x) + 1 from the graph of f(x)
Same shape as f(x) but shifted up 1 unit.
Point
(8, 3)
13) Let f(x) = |𝑥 + 2|
13a) Sketch a graph of f(x)
x
f(x)
point
-4
f(-4) = |−4 + 2|= 2
(-4, 2)
-3
f(-3) = |−3 + 2|= 1
(-3, 1)
-2
f(-2) = |−2 + 2|= 0
(-2, 0)
-1
f(-1) = |−1 + 2| = 1
(-1, 1)
0
f(0) = |0 + 2|= 2
(0, 2)
13b) Find f(-x) = |−𝑥 + 2|
13c) Make a table of values and sketch a graph of f(-x) on the same graph as the graph of f(x)
x
f(x)
point
4
f(4) = |−4 + 2|= 2
(4, 2)
3
f(3) = |−3 + 2|= 1
(3, 1)
2
f(2) = |−2 + 2|= 0
(2, 0)
1
f(1) = |−1 + 2| = 1
(1, 1)
0
f(0) = |0 + 2|= 2
(0, 2)
Graph of f(-x) shown in blue
13d) Describe the transformation to obtain the graph of f(-x) from the graph of f(x)
Same shape as f(x) but reflected over y-axis.
13e) Find -f(x) -f(x) = −|𝑥 + 2|
13f) Make a table of values and sketch a graph of -f(x) on the same graph you have already created
x
f(x)
point
-4
f(-4) = −|−4 + 2|= 2
(-4, -2)
-3
f(-3) =- |−3 + 2|= 1
(-3, -1)
-2
f(-2) =- |−2 + 2|= 0
(-2,- 0)
-1
f(-1) = −|−1 + 2| = 1
(-1, -1)
0
f(0) = −|0 + 2|= 2
(0, -2)
Graph of –f(x) shown in blue
13g) Describe the transformation to obtain the graph of -f(x) from the graph of f(x)
Same shape as f(x) but reflected over x-axis.
15) Let f(x) = (𝑥 − 1)2
15a) Sketch a graph of f(x)
x
f(x)
point
-1
f(-1) = (3-1)2 = 4
(-1, 4)
0
f(0) = (0-1)2 = 1
(0, 1)
1
f(1) = (1-1)2 = 0
(1, 0)
2
f(2) = (2-1)2 = 1
(2, 1)
3
f(3) = (3-1)2 = 4
(3, 4)
15b) Find f(-x) = (-x – 1)2
15c) Make a table of values and sketch a graph of f(-x) on the same graph as the graph of f(x)
x
f(-x)
point
1
f(1) = (-1-1)2 = 4
(1, 4)
0
f(0) = (0-1)2 = 1
(0, 1)
-1
f(-1) = (-(-1)-1)2 = 0
(-1, 0)
-2
f(-2) = (-(-2)-1)2 = 1
(-2, 1)
-3
f(-3) = (-(-3)-1)2 = 4
(-3, 4)
Graph of f(-x) shown in blue
15d) Describe the transformation to obtain the graph of f(-x) from the graph of f(x)
Same shape as f(x) but reflected over the y-axis
15e) Find -f(x) = -f(x) = -(x – 1)2
15f) Make a table of values and sketch a graph of -f(x) on the same graph you have already created
Graph of –f(x) shown in blue
x
f(x)
point
-1
f(-1) = -(3-1)2 = -4
(-1, -4)
0
f(0) = -(0-1)2 = -1
(0, -1)
1
f(1) =- (1-1)2 = 0
(1, 0)
2
f(2) = -(2-1)2 = -1
(2, -1)
3
f(3) = -(3-1)2 = -4
(3, -4)
15g) Describe the transformation to obtain the graph of -f(x) from the graph of f(x)
Same shape as f(x) but reflected over x-axis.
17) Let f(x) = |𝑥|
17a) Sketch a graph of f(x)
x
f(x)
point
-2
𝑓(−2) = |−2| = 2
(-2,2)
-1
𝑓(−1) = |−1| = 1
(-1,1)
0
𝑓(0) = |0| = 0
(0,0)
1
𝑓(1) = |1| = 1
(1,1)
2
𝑓(2) = |2| = 2
(2,2)
17b) Find f(2x) = |2𝑥|
17c) Make a table of values and sketch a graph of f(2x) on the same graph as the graph of f(x)
Graph of f(2x) shown in blue
x
-1
-1/2
0
1/2
1
f(x)
point
𝑓(−1) = |2(−1)| = 2
1
𝑓(−1/2) = |2(− )| = 1
2
𝑓(0) = |2(0)| = 0
1
𝑓(1/2) = |2( )| = 1
2
𝑓(1) = |2(1)| = 2
17d) Describe the transformation to obtain the graph of f(2x) from the graph of f(x)
Same shape as the graph of f(x) but stretched,
it would be equally correct to say the same shape of f(x) but narrower.
(-1,2)
(-1/2, 1)
(0,0)
(1/2, 1)
(1, 2)
1
2
1
2
17e) Find 𝑓 ( 𝑥)= | 𝑥|
1
2
17f) Make a table of values and sketch a graph of 𝑓 ( 𝑥)on the same graph you have already created
1
1
Graph of 𝑓 (2 𝑥)= |2 𝑥| in blue
1
x
𝑓 (2 𝑥)=
point
-4
1
𝑓(−4) = | (−4)| = 2
2
(-4,2)
-2
1
𝑓(−2) = | (−2)| = 1
2
(-2,1)
0
1
𝑓(0) = | (0)| = 0
2
(0,0)
2
1
𝑓(2) = | (2)| = 1
2
(2,1)
4
1
𝑓(4) = | (4)| = 2
2
(4,2)
1
17g) Describe the transformation to obtain the graph of 𝑓 (2 𝑥)from the graph of f(x)
Same shape as f(x) but compressed,
it would also be correct to say same shape as f(x) but wider.
19) Let f(x) = 𝑥 2
19a) Sketch a graph of f(x).
x
f(x)
point
-2
f(-2) = (-2)2 = 4
(-2, 4)
-1
f(-1) = (-1)2 = 1
(-1, 1)
0
f(0) = (0)2 = 0
(0, 0)
1
f(1) = (1)2 = 1
(1, 1)
2
f(2) = (2)2 = 4
(2, 4)
19b) Find f(4x) = (4x)2
19c) Make a table of values and sketch a graph of f(4x)
Graph of f(4x) in blue
x
-2/4
= -1/2
f(x)
point
−1
−1 2
𝑓 ( ) = (4 ∗
) =4
2
2
−1 2
)
4
-1/4
f(-1/4) = (4 ∗
0/4
f(0) = (4*0)2 = 0
=1
−1
( 2 , 4)
(-1/4, 1)
(0, 0)
=0
1/4
2/4
=1/2
1 2
4
f(1/4) = (4 ∗ ) = 1
1
1 2
𝑓 ( ) = (4 ∗ ) = 4
2
2
19d) Describe the transformation to obtain the graph of f(4x) from the graph of f(x)
Same shape as the graph of f(x) but stretched,
it would be equally correct to say the same shape of f(x) but narrower.
(1/4, 1)
(1/2, 4)
1
1
2
19e) Find 𝑓 (4 𝑥) = (4 𝑥)
1
19f) Make a table of values and sketch a graph of 𝑓 (4 𝑥)) on the same graph you have already created
1
1
2
Graph of 𝑓 (4 𝑥) = (4 𝑥) drawn in blue
x
f(x)
point
-8
f(-8) = (4 ∗ −8) = 4
-4
f(-4) =( ∗ −4) = = 1
0
f(0) = (4 ∗ 0) = = 0
4
f(4) = (4 ∗ 4) = = 1
8
f(4) =(4 ∗ 8) = = 4
2
1
2
1
4
2
(0, 0)
1
2
(4, 1)
2
19g) Describe the transformation to obtain the graph of 𝑓 ( 𝑥) from the graph of f(x)
Same shape as f(x) but compressed,
it would also be correct to say same shape as f(x) but wider.
(-4, 1)
1
1
1
4
(-8, 4)
(8, 4)
Section 3.5 Transformation of functions
#21 – 25: Complete the following compressing and stretching problems
21) Let f(x) = |𝑥|
21a) Sketch a graph of f(x)
x
f(x)
point
-2
𝑓(−2) = |−2| = 2
(-2,2)
-1
𝑓(−1) = |−1| = 1
(-1,1)
0
𝑓(0) = |0| = 0
(0,0)
1
𝑓(1) = |1| = 1
(1,1)
2
𝑓(2) = |2| = 2
(2,2)
21b) Find 3f(x) = 3|𝑥|
21c) Make a table of values and sketch a graph of 3f(x) on the same graph as the graph of f(x)
Graph of 3f(x) drawn in blue
x
f(x)
point
-2
𝑓(−2) = 3|−2| = 6
(-2,6)
-1
𝑓(−1) = 3|−1| = 3
(-1,3)
0
𝑓(0) = 3|0| = 0
(0,0)
1
𝑓(1) = 3|1| = 3
(1,3)
2
𝑓(2) = 3|2| = 6
(2,6)
21d) Describe the transformation to obtain the graph of 3f(x) from the graph of f(x)
Same shape as the graph of f(x) but stretched,
it would be equally correct to say the same shape of f(x) but narrower.
1
3
1
3
21e) Find 𝑓(𝑥) = |𝑥|
1
21f) Make a table of values and sketch a graph of 3 𝑓(𝑥) on the same graph you have already created
1
1
Graph of 3 𝑓(𝑥) = 3 |𝑥| drawn in blue
x
f(x)
point
-2
1
𝑓(−2) = |−2| = 2/3
3
(-2,2/3)
-1
1
𝑓(−1) = |−1| = 1/3
3
(-1,1/3)
0
1
𝑓(0) = |0| = 0
3
1
(1,1/3)
1
(2,2/3)
1
𝑓(1) = 3 |1| = 1/3
2
𝑓(2) = 3 |2| = 2/3
1
3
21g) Describe the transformation to obtain the graph of 𝑓(𝑥)from the graph of f(x)
Same shape as f(x) but compressed,
it would also be correct to say same shape as f(x) but wider.
(0,0)
23) Let f(x) = 𝑥 2
a) Sketch a graph of f(x)
x
f(x)
point
-2
f(-2) = (-2)2 = 4
(-2, 4)
-1
f(-1) = (-1)2 = 1
(-1, 1)
0
f(0) = (0)2 = 0
(0, 0)
1
f(1) = (1)2 = 1
(1, 1)
2
f(2) = (2)2 = 4
(2, 4)
23b) Find 3f(x) = 3x2
23c) Make a table of values and sketch a graph of 3f(x) on the same graph as the graph of f(x)
Graph of 3f(x) drawn in blue
x
f(x)
point
-2
f(-2) = 3(-2)2 = 12
(-2, 12)
-1
f(-1) = 3(-1)2 = 3
(-1, 3)
0
f(0) = 3(0)2 = 0
(0, 0)
1
f(1) = 3(1)2 = 3
(1, 3)
2
f(2) = 3(2)2 = 12
(2, 12)
23d) Describe the transformation to obtain the graph of 3f(x) from the graph of f(x)
Same shape as the graph of f(x) but stretched,
it would be equally correct to say the same shape of f(x) but narrower.
1
3
1
3
23e) Find 𝑓(𝑥) = 𝑥 2
1
23f) Make a table of values and sketch a graph of 3 𝑓(𝑥) on the same graph you have already created
1
1
Graph of 3 𝑓(𝑥) = 3 𝑥 2 drawn in blue
x
f(x)
-2
f(-2) = (-2)2 = 4/3
-1
f(-1) = (-1)2 = 1/3
0
f(0) = (0)2 = 0
1
f(1) =3 (1)2 = 1/3
2
f(2) = (2)2 = 4/3
1
3
point
1
3
(-2, 4/3)
1
3
(-1, 1/3)
1
3
(0, 0)
1
(1, 1/3)
1
3
(2, 4/3)
23g) Describe the transformation to obtain the graph of 𝑓(𝑥)from the graph of f(x)
Same shape as f(x) but compressed,
it would also be correct to say same shape as f(x) but wider.
#25 – 50, write the function whose graph is the graph of f(x) = x2, but is
25) Shifted to the left 2 units
f(x+2) = (x+2)2
f(x + #) shifts a graph left. The plus needs to be in a parenthesis.
27) Shifted to the right 5 units f(x - #) shifts a graph left. The minus needs to be in a parenthesis.
f(x – 5) = (x – 5)2
29) Reflected over the x-axis
-f(x) reflects over x-axis
-f(x) = -x2
31) Shifted up 2 units f(x) + # shifts up, the number should not be in a parenthesis
f(x) + 2 = x2 + 2
33) Shifted down 4 units f(x) - # shifts down, the number should not be in a parenthesis
f(x) – 4 = x2 – 4
35) Shifted to the right 2 units and down 3 units
A minus 2 is needed in a parenthesis to shift right
A minus 3 after the parenthesis is needed to shift down.
f(x – 2) – 3 = (x – 2)2 - 3
37) Shifted to the left 2 units and down 3 units
A plus 2 is needed in a parenthesis to shift left
A minus 3 after the parenthesis is needed to shift down.
f(x + 2) – 3 = (x + 2)2 - 3
39) Shifted to the right 2 units and up 3 units
A minus 2 is needed in a parenthesis to shift right
A plus 3 after the parenthesis is needed to shift up.
f(x – 2) + 3 = (x – 2)2 + 3
41) Shifted to the left 2 units and up 3 units
A plus 2 is needed in a parenthesis to shift left
A plus 3 after the parenthesis is needed to shift up.
f(x + 2) + 3 = (x + 2)2 + 3
43) reflected over x-axis and up two units
-f(x) reflects over x-axis
plus 2 after the x2 moves up 2
-f(x) + 2 = -x2 + 2
45) reflected over x-axis and down 2 units
-f(x) reflects over x-axis
minus 2 after the x2 moves down 2
-f(x) - 2 = -x2 - 2
47) reflected over x-axis and right 3 units
-f(x) reflects over x-axis
A minus 3 is needed in a parenthesis to shift right
-f(x – 3) = -(x – 3)2
49) reflected over x-axis and left 2 units
-f(x) reflects over x-axis
A plus is needed in a parenthesis to shift left
-f(x + 2) = -(x+2)2
#51 – 56: Use the graph of f(x) below to sketch a graph and describe the transformation.
51)
51a) f(x – 2)
Each point needs to be shifted 2 units to the right.
graph of f(x -2) is drawn in blue.
51b) f(x + 1)
Each point needs to be moved 1 unit to the left.
51c) f(x) – 2
Each point needs to be shifted down 2 units
51d) f(x) + 1 Each point needs to be shifted up 1 unit.
51e) f(x-2) + 1
Each point needs to be shifted 2 right and up 1
51f) f(x + 1) – 2 Each point needs to be shifted left 1 and down 2
51g) f(-x)
each point needs to be reflected over the y-axis
51h) –f(x) Each point needs to be reflected over the x-axis
53)
53a) f(x – 1) shift each point right 1
53b) f(x + 2) Shift each point left 2
53c) f(x) – 1 Shift each point down 1
53d) f(x) + 2 Shift each point up 2
53e) f(x-1) + 2 Shift each point right 1 and up 2
53f) f(x + 2) – 1
Shift each point left 2 and down 1
53g) f(-x) reflect each point over the y-axis
The graph is already symmetric to the y-axis so the graph looks the same as when it started. Original
graph is in red, f(-x) is in blue.
53h) –f(x) reflect each point over the x-axis
55)
55a) f(x – 2)
each point must be shifted right 2
55b) f(x + 2)
each point must be shifted left 2
55c) f(x) – 3
each point must be moved down 3
55d) f(x) + 3
each point must be shifted up 3
55e) f(x-2) + 3
each point must be shifted right 2 and up 3
55f) f(x + 2) – 3
each point must be shifted left 2 and down 3
55g) f(-x)
each point must be reflected over the y-axis
55h) –f(x)
each point must be reflected over the x-axis