Vertical Translations: [ Interactive Graph ]
If k is any positive real number then,
− 3 by shifting all points down 3 units.
Horizontal Translations: [ Interactive Graph ]
− 3) by shifting all points right 3 units.
Interactive Graph ]
For the first function f(x) = −sqrt(x) all of the y-values are negative which results in a reflection about the x-axis. For the second function f(x) = sqrt(−x) all of the x-values must be negative thus resulting in a reflection about the y-axis.
Sketch the graph.
The -1 indicates a reflection of the graph of the squaring function f(x) = x^2 about the x-axis. Be sure to graph the squaring function using a dashed curve because it will be used as a guide and is not the answer. Next, reflect all points about the x-axis and draw in the final graph with a solid curve.
General Steps for Graphing Functions using Transformations:
1. Identify and graph the basic function using a dashed curve.
2. Identify any reflections first and sketch them using the basic function as a guide.
3. Identify any translations.
4. Use this information to sketch the final graph using a solid curve.
Graph the function and determine the domain and range.
y = sqrt(x)Next, notice the reflection about the y-axis,
y = sqrt(−x)And finally, we see a shift up 1 unit.
y = sqrt(−x) + 1
y = x^2Next, notice the shift right 3 units,
y = (x − 3)^2And finally, we see a shift down 2 units.
y = (x − 3)^2 − 2
y = abs(x)Next, notice the reflection about the x-axis,
y = −abs(x)Finally we see a shift left 1 unit and down 2 units
y = −abs(x+1) − 2
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