5A - Exploring transformations
To explore transformations of graphs, we will first look at the case study of transforming the parabola \(y=x^2\). Throughout this section we will be referring to the original graph (\(y\)) which is transformed to give the image graph (\(y_1\)).
Translations:
A translation moves ("slides") the graph left/right/up/down.
Horizontal translations: \((x,y)\rightarrow (x+h,y)\) is a translation of \(h\) units parallel to the \(x\)-axis.
A translation moves ("slides") the graph left/right/up/down.
Horizontal translations: \((x,y)\rightarrow (x+h,y)\) is a translation of \(h\) units parallel to the \(x\)-axis.
- If \(h>0\) (positive) the graph is translated in the positive \(x\)-direction.
- If \(h<0\) (negative) the graph is translated in the negative \(x\)-direction.
- If \(k>0\) (positive) the graph is translated in the positive \(y\)-direction.
- If \(k<0\) (negative) the graph is translated in the negative \(y\)-direction.
5A - Example 1: Translations of a parabola
The parabola \(y=x^2\) is translated 2 units in positive \(x\)-direction and 1 unit in the positive \(y\)-direction.
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5A - Example 1: Video solution
5A - Example 1: Practice
Question 1: ABC Question 2: ABC 5A - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Dilations:
A dilation "stretches" or "compresses" as graph towards or away from a specified axis.
Dilations from the \(x\)-axis: \((x,y)\rightarrow (x,ay)\) is a dilation by a factor of \(a\) from the \(x\)-axis.
A dilation "stretches" or "compresses" as graph towards or away from a specified axis.
Dilations from the \(x\)-axis: \((x,y)\rightarrow (x,ay)\) is a dilation by a factor of \(a\) from the \(x\)-axis.
- If \(0<a<1\) the graph is "compressed" (towards the \(x\)-axis).
- If \(a>1\) the graph is "stretched" (away from the \(x\)-axis).
- If \(0<n<1\) the graph is "compressed" (towards the \(y\)-axis).
- If \(n>1\) the graph is "stretched" (away from the \(y\)-axis).
5A - Example 2: Dilating a parabola
The parabola \(y=x^2\) is dilated by a factor of \(\frac{1}{2}\) from the \(x\)-axis.
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5A - Example 2: Video solution
5A - Example 2: Practice
Question 1: ABC Question 2: ABC 5A - Example 2: Solutions
Question 1: ABC Question 2: ABC |
Reflections:
A reflection "flips" a graph over a specified axis.
Reflections over an axis:
A reflection "flips" a graph over a specified axis.
Reflections over an axis:
- \((x,y)\rightarrow (x,-y)\) reflects the graph over the \(x\)-axis.
- \((x,y)\rightarrow (-x,y)\) reflects the graph over the \(y\)-axis.
5A - Example 3: Reflecting a parabola
Consider the parabola \(y=(x_2)^2\). The original graph is dotted onto the set of axes below.
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5A - Example 3: Video solution
5A - Example 3: Practice
Question 1: ABC Question 2: ABC 5A - Example 3: Solutions
Question 1: ABC Question 2: ABC |
Combining transformations:
When combining transformations:
When combining transformations:
- The order of transformations is very important!
- Only carry out one transformations (dilations, reflection or translation) in each set of brackets.
- Make sure you carry out the transformations in the order as stated in the question.
5A - Example 4: Combining transformations to transform a parabola
■ What do you notice about the rules and the graphs of the two image graphs, \(y_1\) and \(y_2\)? |
5A - Example 4: Video solution
5A - Example 4: Practice
Question 1: ABC Question 2: ABC 5A - Example 4: Solutions
Question 1: ABC Question 2: ABC |