Transformations of Functions - Verulam.S3.Amazonaws
Transformations of Functions
Transformation of functions f(x+c) move left by c in x axis direction f(x-c) move right by c in x axis direction f(x) + c move up by c in y axis direction f(x) –c move down by c in y axis direction cf(x) stretch y axis by factor c f(cx) stretch in x axis by factor 1/c (keeping y axis fixed )
Vertical Shifts Let f be a function and c a positive real number. •The graph of y = f (x) + c is the graph of y = f (x) shifted c units vertically upward. •The graph of y = f (x) – c is the graph of y = f (x) shifted c units vertically downward. y = f (x) + c y = f (x) c c y = f (x)
y = f (x) - c
Example • Use the graph of y = x2 to obtain the graph of y = x2 + 4.
10 8 6 4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example cont.
• Use the graph of y = x2 to obtain the graph of y = x2 + 4. Step 1 Graph f (x) = x2. The graph of the standard quadratic function is shown. Step 2 Graph g(x) = x2+4. Because we add 4 to each value of x2 in the range, we shift the graph of f vertically 4 units up.
10 8 6 4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Horizontal Shifts Let f be a function and c a positive real number. •The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units. •The graph of y = f (x - c) is the graph of y = f (x) shifted to the right c units. y = f (x)
y = f (x + c)
c
y = f (x)
c
y = f (x - c)
Text Example Use the graph of f (x) to obtain the graph of h(x) = (x + 1)2 – 3. Solution 5
x2.
Step 1 Graph f (x) = The graph of the standard quadratic function is shown.
4 3 2 1 -5 -4 -3 -2
1)2.
-1 -1 -2
1
Step 2 Graph g(x) = (x + Because we -3 -4 add 1 to each value of x in the -5 domain, we shift the graph of f horizontally one unit to the left. Step 3 Graph h(x) = (x + 1)2 – 3. Because we subtract 3, we shift the graph vertically down 3 units.
2
3 4
5
Reflection about the x-Axis • The graph of y = - f (x) is the graph of y = f (x) reflected about the x-axis.
Reflection about the y-Axis • The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis.
Stretching and Shrinking Graphs Let f be a function and c a positive real number. •If c > 1, the graph of y = c f (x) is the graph of y = f (x) vertically stretched by multiplying each of its y-coordinates by c. •If 0 < c < 1, the graph of y = c f (x) is the graph of y = f (x) vertically shrunk by multiplying each of its y-coordinates by c. g(x) = 2x2
f (x) = x2
10 9 8 7 6 5 4 3 2 1 -4 -3 -2 -1
h(x) =1/2x2
1
2
3
4
Sequence of Transformations A function involving more than one transformation can be graphed by performing transformations in the following order. 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflecting 4. Vertical shifting
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 1: Because x is replaced with x+3, the graph is shifted 3 units to the left.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 2: Because the equation is not multiplied by a constant, no stretching or shrinking is involved.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 3: Because x remains as x, no reflecting is involved.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 10
Solution:
8 6
Step 4: Because 4 is subtracted, shift the graph down 4 units.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Transformation of functions f(x+c) move left by c in x axis direction f(x-c) move right by c in x axis direction f(x) + c move up by c in y axis direction f(x) –c move down by c in y axis direction cf(x) stretch y axis by factor c f(cx) stretch in x axis by factor 1/c (keeping y axis fixed )
Vertical Shifts Let f be a function and c a positive real number. •The graph of y = f (x) + c is the graph of y = f (x) shifted c units vertically upward. •The graph of y = f (x) – c is the graph of y = f (x) shifted c units vertically downward. y = f (x) + c y = f (x) c c y = f (x)
y = f (x) - c
Example • Use the graph of y = x2 to obtain the graph of y = x2 + 4.
10 8 6 4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example cont.
• Use the graph of y = x2 to obtain the graph of y = x2 + 4. Step 1 Graph f (x) = x2. The graph of the standard quadratic function is shown. Step 2 Graph g(x) = x2+4. Because we add 4 to each value of x2 in the range, we shift the graph of f vertically 4 units up.
10 8 6 4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Horizontal Shifts Let f be a function and c a positive real number. •The graph of y = f (x + c) is the graph of y = f (x) shifted to the left c units. •The graph of y = f (x - c) is the graph of y = f (x) shifted to the right c units. y = f (x)
y = f (x + c)
c
y = f (x)
c
y = f (x - c)
Text Example Use the graph of f (x) to obtain the graph of h(x) = (x + 1)2 – 3. Solution 5
x2.
Step 1 Graph f (x) = The graph of the standard quadratic function is shown.
4 3 2 1 -5 -4 -3 -2
1)2.
-1 -1 -2
1
Step 2 Graph g(x) = (x + Because we -3 -4 add 1 to each value of x in the -5 domain, we shift the graph of f horizontally one unit to the left. Step 3 Graph h(x) = (x + 1)2 – 3. Because we subtract 3, we shift the graph vertically down 3 units.
2
3 4
5
Reflection about the x-Axis • The graph of y = - f (x) is the graph of y = f (x) reflected about the x-axis.
Reflection about the y-Axis • The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis.
Stretching and Shrinking Graphs Let f be a function and c a positive real number. •If c > 1, the graph of y = c f (x) is the graph of y = f (x) vertically stretched by multiplying each of its y-coordinates by c. •If 0 < c < 1, the graph of y = c f (x) is the graph of y = f (x) vertically shrunk by multiplying each of its y-coordinates by c. g(x) = 2x2
f (x) = x2
10 9 8 7 6 5 4 3 2 1 -4 -3 -2 -1
h(x) =1/2x2
1
2
3
4
Sequence of Transformations A function involving more than one transformation can be graphed by performing transformations in the following order. 1. Horizontal shifting 2. Vertical stretching or shrinking 3. Reflecting 4. Vertical shifting
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 1: Because x is replaced with x+3, the graph is shifted 3 units to the left.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 2: Because the equation is not multiplied by a constant, no stretching or shrinking is involved.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 Solution:
10 8 6
Step 3: Because x remains as x, no reflecting is involved.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
Example • Use the graph of f(x) = x3 to graph g(x) = (x+3)3 - 4 10
Solution:
8 6
Step 4: Because 4 is subtracted, shift the graph down 4 units.
4 2
-10
-8
-6
-4
-2
2 -2 -4 -6 -8
-10
4
6
8
10
