Can you arrange the functions \(f(x)=x^2\), \(f(x)=-1\), \(f(x)=x\) and \(f(x)=x^3\) in the table below so that they have the properties stated in each cell?
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Graph is straight \(y\)-axis is a line of symmetry \[f(x)= \cdots\] |
Graph is straight \(y\)-axis is not a line of symmetry \[f(x)= \cdots\] |
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Graph is not straight \(y\)-axis is a line of symmetry \[f(x)= \cdots\] |
Graph is not straight \(y\)-axis is not a line of symmetry \[f(x)= \cdots\] |
Can you explain why each function is in each cell?
We need to think about what is meant by the two properties. They both describe properties of the graph of a function, but can be interpreted algebraically too. For example, the \(y\)-axis is a line of symmetry if the graph would look the same when reflected in the \(y\)- axis, but what would this mean for the input and output values of the function?
The only way of arranging the four given functions in the table is shown here.
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Graph is straight \(y\)-axis is a line of symmetry \[f(x)=1\] |
Graph is straight \(y\)-axis is not a line of symmetry \[f(x)=x\] |
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Graph is not straight \(y\)-axis is a line of symmetry \[f(x)=x^2\] |
Graph is not straight \(y\)-axis is not a line of symmetry \[f(x)=x^3\] |
There are several ways to explain why these functions have the properties in the corresponding row and column. Here are some suggestions.
The graph of \(f(x)=-1\) has gradient \(0\), so it is parallel to the \(x\)-axis. Therefore it is straight and the \(y\)-axis is a line of symmetry. Another way to see the symmetry is to note that all input values of \(x\) give the same output value, so in particular, \(f(2)=f(-2)\), \(f(3)=f(-3)\), and in general \(f(x)=f(-x).\) Therefore the function behaves the same way on either side of the \(y\)-axis.
The graph of \(f(x)=x\) has gradient 1, so it is straight, but it does not look the same when reflected in the \(y\)-axis.
The graph of \(f(x)=x^2\) gets steeper and steeper as \(x\)-values move further away from \(0\), so it is not straight, but \((-x)^2=x^2\), so the \(y\)-axis is a line of symmetry.
The graph of \(f(x)=x^3\) gets steeper and steeper as \(x\)-values move further away from \(0\), so it is not straight. However, \(f(-2)=(-2)^3=-8,\) and \(f(2)=2^3=8\), so the graph of \(f(x)=x^3\) does not look the same when reflected in the \(y\)-axis.
Which other functions could fit in the cells of this table?
Which straight lines are symmetric about the \(y\)-axis? Are all straight lines the graphs of functions of \(x\)?
The line \(x=0\) is straight and symmetric about the \(y\)-axis, but a function of \(x\) has to give a single output for any valid input value of \(x\). If a line such as \(x=3\) were the graph of a function of \(x\), what could the output value, \(f(3),\) be?
For the other cells, you might find it helpful to make a list of the functions you know about and try to sketch their graphs. Here is an alternative way to complete this function square.
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Graph is straight \(y\)-axis is a line of symmetry \[f(x)=1\] |
Graph is straight \(y\)-axis is not a line of symmetry \[f(x)=-x\] |
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Graph is not straight \(y\)-axis is a line of symmetry \[f(x)=x^2+1\] |
Graph is not straight \(y\)-axis is not a line of symmetry \[f(x)=\frac{1}{x} \text{ for } x\neq 0\] |
We have to specify that \(x\) is not zero when we write \(f(x)=\dfrac{1}{x}\) because \(\dfrac{1}{x}\) isn’t defined if \(x=0.\) It is important to think about which inputs are valid when you write down a formula for a function.
