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Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

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If you plot these graphs they may look the same, but are they?

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Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

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The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

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Quadratic graphs are very familiar, but what patterns can you explore with cubics?

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This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

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This problem challenges you to find cubic equations which satisfy different conditions.

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This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

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The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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Sketch the graphs for this implicitly defined family of functions.

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Can you work out the equations of the trig graphs I used to make my pattern?

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In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

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The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

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Which curve is which, and how would you plan a route to pass between them?

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This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

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Can you massage the parameters of these curves to make them match as closely as possible?

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Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

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Plot the graph of x^y = y^x in the first quadrant and explain its properties.

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Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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Compares the size of functions f(n) for large values of n.

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Can you draw the height-time chart as this complicated vessel fills with water?

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Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

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Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

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Can you work out which processes are represented by the graphs?

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How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.

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This problem challenges you to sketch curves with different properties.

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Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

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Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

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Explore how can changing the axes for a plot of an equation can lead to different shaped graphs emerging

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Various solids are lowered into a beaker of water. How does the water level rise in each case?

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Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

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Explore the rates of growth of the sorts of simple polynomials often used in mathematical modelling.

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Which line graph, equations and physical processes go together?

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10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

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By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

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Can you sketch these difficult curves, which have uses in mathematical modelling?

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Can you match the charts of these functions to the charts of their integrals?