Solution 2.3:8c

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By completing the square, we can rewrite the function as
By completing the square, we can rewrite the function as
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{{Displayed math||<math>f(x) = x^{2}-6x+11 = (x-3)^{2} - 3^{2} + 11 = (x-3)^{2} + 2,</math>}}
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<math>f\left( x \right)=x^{2}-6x+11=\left( x-3 \right)^{2}-3^{2}+11=\left( x-3 \right)^{2}+2,</math>
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and when the function is written in this way, we see that the graph <math>y = (x-3)^{2} + 2</math> is the same curve as the parabola <math>y=x^{2}</math>, but shifted two units up and three units to the right (see sub-exercise a and b).
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and when the function is written in this way, we can see that the graph
 
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<math>y=\left( x-3 \right)^{2}+2</math>
 
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is the same curve as the parabola
 
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<math>y=x^{2}</math>, but shifted two units up and three units to the right (see sub-exercise d and e).
 
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{| align="center"
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[[Image:2_3_8_c.gif|center]]
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|align="center"|[[Image:2_3_8_c-1.gif|center]]
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||&nbsp;
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|align="center"|[[Image:2_3_8_c-2.gif|center]]
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|-
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|align="center"|<small>The graph of ''f''(''x'')&nbsp;=&nbsp;''x''²</small>
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||
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|align="center"|<small>The graph of ''f''(''x'')&nbsp;=&nbsp;''x''²&nbsp;-&nbsp;6x&nbsp;+&nbsp;11</small>
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|}

Current revision

By completing the square, we can rewrite the function as

\displaystyle f(x) = x^{2}-6x+11 = (x-3)^{2} - 3^{2} + 11 = (x-3)^{2} + 2,

and when the function is written in this way, we see that the graph \displaystyle y = (x-3)^{2} + 2 is the same curve as the parabola \displaystyle y=x^{2}, but shifted two units up and three units to the right (see sub-exercise a and b).


 
The graph of f(x) = x² The graph of f(x) = x² - 6x + 11
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