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<mdoxml version="1.0"><p>Steve wishes to draw a quadratic polynomial $y(x) = ax^2+bx+c$ through the three points $(x, y)=(1,4), (2, 5), (3, 7)$. He writes down this expression:</p>
<p>$$y(x) = 4\frac{(x-2)(x-3)}{(1-2)(1-3)}+5\frac{(x-1)(x-3)}{(2-1)(2-3)}+7\frac{(x-1)(x-2)}{(3-1)(3-2)}$$</p>
<p>Does this solve the problem that Steve was trying to address?</p>
<p> </p>
<p>Using Steve's example as a guide, can you construct a quadratic polynomial which passes through the three points $(1,2), (2, 4), (4, -1)$? Is this the only such quadratic polynomial?</p>
<p> </p>
<p>Can you construct a cubic polynomial which passes through the four points $(1,2), (2, 4), (3, 7),(4, -1)$. Is this the only such cubic polynomial?</p>
<p> </p>
<p>By this stage, the particular examples you have constructed should give you ideas about how to construct a 'general' case. Use your insights to answer these questions:</p>
<p> </p>
<p>Can you write down an expression for a line passing through the two points $(2, 7)$ and $(8,-6)$ using this method?</p>
<p> </p>
<p>Can you always fit a quadratic polynomial through three points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$?</p>
<p> </p>
<p>Can you always fit a quartic polynomial through five points $(x_i, y_i)$ ($i=1\dots 5$) where exactly two of $x_i$ are zero?</p>
<p> </p>
<p>How many different polynomials can you construct which would pass through the points $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)$?</p>
<p> </p>
<p>Explore sets of points through which it is not possible to fit a polynomial.</p>
<p> </p>
<p><strong>Extension</strong>: For various numbers of points and degrees of polynomial you might wish to consider when the fitting is unique, when it is possible with multiple polynomials and when it is impossible.</p></mdoxml>
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<mdoxml version="1.0"><p><span class="editorial">This problem didn't receive a solution when it was published in March 2012. Perhaps you can submit a solution?</span></p></mdoxml>
<mdoxml version="1.0"><h3>Why do this problem?</h3>
<p>This problem introduces students to interpolation and the concept of 'building' algebraic solutions to problems. The result is very interesting mathematically. It is based upon the idea of a 'generic example': a particular example which encapsulates in a clear way all of the properties of a more general case. The ideas in this problem pave the way for patterns of thinking which are to be found
in university mathematics courses, and the concept of interpolation is valuable in both mathematics and science. This problem also raises the idea from proof that constructing an example with the correct properties proves that an example exists, whereas inability to construct an example with the correct properties does not necessarily prove that such an example doesn't exist.</p>
<p> </p>
<h3>Possible approach</h3>
<div>This problem requires that students understand the concept of fitting a curve through a set of points to mean, roughly, 'write down a function $y=f(x)$ which passes <em>exactly</em> through each of the specified points', and will need to understand that we are not interested in the function anywhere <em>except</em> at these isolated points.</div>
<div> </div>
<div> </div>
<div>Once this is understood you can simply put the first part of the question on the board and pose the question 'does this solve Steve's problem?'. The algebraic expression might initially appear intimidating or mysterious, but gradually students will realise that most of the expression evaluates to zero at each of the points and then will realise why the numbers have been written as they have
been.</div>
<div> </div>
<div>It is important that students be given enough time to grapple with the first algebraic expression as by doing this they will appreciate the structure which leads to it 'obviously' solving the problem. You can do this in small groups or as a whole class-based discussion (you can use the Key Questions to help with this discussion). Be sure to do this <em>without</em> pencil and paper in the
first instance, because some students might be tempted to expand out all of the brackets and this destroys the structure of the construction.</div>
<div> </div>
<div>Once this expression is seen to solve Steve's problem, a brief discussion can be had concerning why Steve wrote the expression as he did. For example, why is the whole expression not expanded? Why not simplify some of the brackets such as $(1-2)$?</div>
<div> </div>
<div>Now students should be ready to tackle the rest of the problem on their own. Clear critical thinking is required, and if students believe that certain results are 'obvious' be sure to get them to explain clearly what is so obvious: students will need to pay particular attention to zeros occurring in their construction.</div>
<div> </div>
<div>For the final part it is worth asking students to consider: does a quadratic fit through the five points? How do you know (if you claim that you do) either way?</div>
<div> </div>
<div>Finally, some students might really want to simplify the expressions to find the coefficients of the polynomials or to plot the graphs to see the shapes of the answers. This is fine if they want to!</div>
<div> </div>
<h3>Key questions</h3>
<p>What does Steve's expression evaluate to at each of the three points?</p>
<p>Why has Steve not expanded the brackets or simplified the numbers?</p>
<p>Do you think that Steve's construction could be generalised?</p>
<p>Under which circumstances would Steve's construction break down?</p>
<p>How many degrees of freedom does a quadratic polynomial have?</p>
<p> </p>
<h3>Possible extension</h3>
<p>There are some extension possibilities in the question. Other extension possibilities are to use a spreadsheet or computer to produce the fitting polynomials for $4$ or $5$ variable points. This is mathematically very interesting and will lead to the understanding that the fitting polynomials, whilst exactly hitting all of the points, are often very unstable and with wildly varying shape -
this is a good open investigation which might arise from this problem.</p>
<p> </p>
<h3>Possible support</h3>
<p>The main idea in this question concerns constructing a fitting polynomial by arranging brackets and coefficients in a sensible, organised manner. This key idea can be practised by looking at fitting quadratics through 3 points: give a few triples of points and have students construct the quadratics which go through these. <br></br>
<br></br>
<br></br>
</p></mdoxml>
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<mdoxml version="1.0"><p>With an interpolation, we are only interested in the properties of the function at the specified interpolation points, so there is no need to worry about the form of the functions elsewhere or to sketch them (although feel free to explore this if you are interested!).</p>
<p> </p>
<p>Don't be tempted to expand out Steve's expression, as this will hide the properties of the construction which allow you to see generalisations.</p>
<p> </p>
<p>To get started, try substituting in a point and see what cancels or vanishes!</p></mdoxml>
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Polynomial interpolation
Can you fit polynomials through these points?
Advanced Algebra
Polynomials
Advanced Algebra
Roots of polynomial equations
Algebra
Creating expressions/formulae
Advanced Algebra
Interpolation
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