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The following linear graph represents a function 𝑔 of 𝑥 after a reflection in the 𝑦-axis.
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Find the original function 𝑓 of 𝑥.
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There are a few ways to approach this problem.
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One way would be to find the equation of the function 𝑔 of 𝑥 given on the graph.
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As it is a linear function, we know it can be written in the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept.
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It is clear from the graph that the 𝑦-intercept is negative, four.
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We know that the slope or gradient of any line is equal to the rise over the run.
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Choosing the two points negative four, four and negative two, zero which lie on the line, the slope is equal to negative four over two.
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This is equal to negative two.
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The function 𝑔 of 𝑥 has equation negative two 𝑥 minus four.
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We are told that 𝑔 of 𝑥 is a reflection of 𝑓 of 𝑥 in the 𝑦-axis.
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Therefore, 𝑓 of 𝑥 is also a reflection of 𝑔 of 𝑥 in the 𝑦-axis.
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This means that 𝑓 of 𝑥 is equal to 𝑔 of negative 𝑥.
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The function 𝑓 of 𝑥 is therefore equal to negative two multiplied by negative 𝑥 minus four.
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This simplifies to two 𝑥 minus four.
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The original linear function 𝑓 of 𝑥 has equation two 𝑥 minus four.
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An alternative method would have been to have sketched the reflection of 𝑔 of 𝑥 on the graph.
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When reflecting a graph in the 𝑦-axis, we know that the 𝑦-intercept remains the same and the roots change signs.
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This means that 𝑓 of 𝑥 will still intercept the 𝑦-axis at negative four and intercept the 𝑥-axis at positive two.
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As the slope or gradient of this line is two and the 𝑦-intercept is negative four, the equation of the line is two 𝑥 minus four.
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This confirms the answer we found using our first method.