5G - Determining the transformations that have occurred
The language of transformations:
In VCE mathematical methods, the language used to describe transformations is very important.
In VCE mathematical methods, the language used to describe transformations is very important.
- Dilations are expresses as “a factor of \(a\) from the \(x\)-axis” or “by a factor of \(\frac{1}{n}\) from the \(y\)-axis”.
- Reflections are expressed as “a reflection over the \(x\)-axis” or “a reflection over the \(y\)-axis”.
- Translation are expressed as “a translation of \(h\) units in the positive/negative \(x\)-direction”; or, “a translation of \(k\) units in the positive/negative \(y\)-direction”.
Determining the transformations directly from the rule:
In the previous examples in this section we have determined the transformations of different graphs by inspecting the rule directly. \[y=a\times f(n(x-h))+k\] For the general rule above, we know that:
In the previous examples in this section we have determined the transformations of different graphs by inspecting the rule directly. \[y=a\times f(n(x-h))+k\] For the general rule above, we know that:
- \(a\) causes a dilation by a factor of \(a\) units from the \(x\)-axis. If \(a<0\) the graph is reflected over the \(x\)-axis.
- \(n\) causes a dilation by a factor of \(\frac{1}{n}\) units from the \(y\)-axis. If \(n<0\) the graph is reflected over the \(y\)-axis.
- \(h\) causes a translation of \(h\) units parallel to the \(x\)-axis.
- \(k\) causes a translation of \(k\) units parallel to the \(y\)-axis.
Determining the transformations using reverse mapping notation:
We can also determine the transformations that have occurred using reverse mapping notation. To determine the transformations that have occurred to the original equation to produce the image graph:
We can also determine the transformations that have occurred using reverse mapping notation. To determine the transformations that have occurred to the original equation to produce the image graph:
- Reintroduce the dashes and rearrange the equation to resemble the original (basic) equation.
- Equate \(y=\) and \(x=\) with the relevant parts of the rearranged equation (now with dashes).
- Solve for the dashed terms in the equations found in Step 2, and express as a sequence “maps”.
- Interpret the mapping notation found in Step 3 as a worded description of the transformations.
5G - Example 1: Determining the transformations that have occurred
State, in order, the sequence of transformation that have occurred to \(y=\frac{1}{x}\) to give \[y_1=\frac{-1}{x-3}+2\]
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5G - Example 1: Video solution
5G - Example 1: Practice
Question 1: Determine a sequence of transformations that will map \(y=\frac{1}{x}\) to \(y=\frac{-3}{x-5}\). Question 2: Determine a sequence of transformations that will map \(y=x^2\) to \(y=2x^2+3\). 5G - Example 1: Solutions
Question 1:
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5G - Example 2: Determining the transformations that have occurred
State, in order, the sequence of transformation that have occurred to \(y=\frac{1}{x^2}\) to give \[y_1=\frac{2}{(x+5)^2}-4\]
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5G - Example 2: Video solution
5G - Example 2: Practice
Question 1: Determine a sequence of transformations that will map \(y=\frac{1}{x^2}\) to \(y=\frac{-1}{(x+6)^2}+16\). Question 2: ABC 5G - Example 2: Solutions
Question 1:
ABC |
5G - Example 3: Determining the transformations that have occurred
State, in order, the sequence of transformation that have occurred to \(y=\sqrt{x}\) to give \[y_1=\sqrt{5-2x}+3\]
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5G - Example 3: Video solution
5G - Example 3: Practice
Question 1: ABC Question 2: ABC 5G - Example 3: Solutions
Question 1: ABC Question 2: ABC |
5G - Example 4: Determining the transformations that have occurred
State, in order, the sequence of transformation that have occurred to \(x^2+y^2=16\) to give the image graph \[(x+2)^2+(y-6)^2=16\]
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5G - Example 4: Video solution
5G - Example 4: Practice
Question 1: ABC Question 2: ABC 5G - Example 4: Solutions
Question 1: ABC Question 2: ABC |
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Additional Exercises
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Topic Worksheets
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Other Resources
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Determine a sequence of transformations that will map \(y=2\sqrt{x}+4\) to \(y=3\sqrt{x+1}-12\).
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Solutions:
Question 1: Dilation by a factor of \(\frac{3}{2}\) from the \(x\)-axis, followed by a translation of \(1\) unit in the negative \(x\)-direction and \(18\) units in the negative \(y\)-direction. |