s ection 1.7 transformations. t ransformation c ategories the transformations fall into three broad...
TRANSCRIPT
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SECTION 1.7Transformations
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TRANSFORMATION CATEGORIES
The transformations fall into three broad categories: 1. Shifts2. Reflections3. Scalings
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SHIFT UP
Graph f(x) = √x Graph g(x)= f(x) + 2
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VERTICAL SHIFTS
Suppose f is a function and k is a positive number
To graph y=f(x) + k, shift the graph of y = f(x) up k units by adding k to the y-coordinates of the points on the graph of f
To graph y=f(x) - k, shift the graph of y = f(x) down k units by subtracting k from the y-coordinates of the points on the graph of f
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ANOTHER TRANSFORMATION
Graph f(x) = √x Graph g(x) = f(x+2) What happens when we add 2 to the input of
the function?
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HORIZONTAL SHIFTS
Suppose f is a function and h is a positive number
To graph y = f(x + h), shift the graph of y=f(x) left h units by subtracting h from the x-coordinates of the points on the graph of f
To graph y=f(x - h), shift the graph of y = f(x) right h units by adding h to the x-coordinates of the points on the graph of f
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HORIZONTAL AND VERTICAL CHANGE
Changes to the outputs from a function affect the y-coordinates of the graph, resulting in some kind of vertical change
Changes to the inputs to a function affect the x-coordinates of the graph, resulting in some kind of horizontal change
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EXAMPLE
1. Graph . Plot at least three points.
2. Use your graph in 1 to graph
3. Use your graph in 1 to graph
4. Use your graph in 1 to graph
xxf )(
1)( xxg
1)( xxj
23)( xxm
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SOLUTION
1.
2.
3.
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SOLUTION (CONTINUED)
4.
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REFLECTIONS
Suppose f is a function
To graph y = -f(x), reflect the graph of y = f(x) across the x-axis by multiplying the y-coordinates of the points on the graph of f by -1
To graph y = f(-x), reflect the graph of y = f(x) across the y-axis by multiplying the x-coordinates of the points on the graph of f by -1
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EXAMPLE
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EXAMPLE
Let . Use the graph of f to graph the following functions below. Also, state their domains and ranges.
1.
2.
3.
Order of transformation is very important!
xxf )(
xxf )(
xxj 3)(
xxm 3)(
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SOLUTION
1.
2.
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SOLUTION (CONTINUED)
3.
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SCALING
Let f(x) = √x Graph g(x) = 2f(x) Graph h(x) = ½ f(x)
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VERTICAL SCALINGS
Suppose f is a function and a > 0 To graph y = a f(x), multiply all of the
y-coordinates of the points on the graph of f by a
We say the graph of f has been vertically scaled by a factor of a If a > 1, we say the graph of f has undergone a
vertical stretch (expansion, dilation) by a factor of a
If 0 < a < 1, we say the graph of f has undergone a vertical shrink (compression, contraction) by a factor of 1/a
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LAST TRANSFORMATION
Let f(x) = √x Graph g(x) = f(2x) Graph h(x) = f( ½ x)
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HORIZONTAL SCALINGS
Suppose f is a function and b > 0 To graph y = f(bx), divide all of the x-
coordinates of the points on the graph of f by b
We say the graph of f has been horizontally scaled by a factor of 1/b If 0 < b < 1, we say the graph of f has
undergone a horizontal stretch (expansion, dilation) by a factor of 1/b
If b > 1, we say the graph of f has undergone a vertical shrink (compression, contraction) by a factor of b
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ORDER OF TRANSFORMATIONS Suppose f is a function. To graph
Rearrange the formula so that it has the above from1. Subtract H from each of the x-coordinates of the points
on the graph of f. This results in a horizontal shift to the• left if H > 0 • right if H< 0
2. Divide the x-coordinates of the points on the graph obtained in Step 1 by B. This results in a horizontal scaling, but may also include a reflection about the y-axis if B < 0.
3. Multiply the y-coordinates of the points on the graph obtained in Step 2 by A. This results in a vertical scaling, but may also include a reflection about the x-axis if A < 0
4. Add K to each of the y-coordinates of the points on the graph obtained in Step 3. This results in a vertical shift • up if K > 0 • down if K< 0.
KHBxAfxg )()(
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EXAMPLE
Let . Use the graph of f to graph the following functions below. Also, state their domains and ranges.
1.
2.
3.
xxf )(
xxg 3)(
xxj 9)(
2
31)(
x
xm
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SOLUTION
1.
2.
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SOLUTION (CONTINUED)
3.
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SOLUTION (CONTINUED)
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EXAMPLE
Below is the complete graph of y = f(x) Use it to graph
2
)21(34)(
xfxg
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SOLUTION
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EXAMPLE
Let f(x) = x2
Find and simplify the formula of the function g(x) whose graph is the result of f undergoing the following sequence of transformations
Check your answer using a graphing calculator1. Vertical shift up 2 units2. Reflection across the x-axis3. Horizontal shift right 1 unit4. Horizontal stretch by a factor of 2
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SOLUTION
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SOLUTION (CONTINUED)