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#### Resources tagged with Graphs similar to Three Ways:

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Broad Topics > Sequences, Functions and Graphs > Graphs

### Three Ways

##### Stage: 5 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

### Without Calculus

##### Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

### Parabolas Again

##### Stage: 4 and 5 Challenge Level:

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?

### Mathsjam Jars

##### Stage: 4 Challenge Level:

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

### More Parabolic Patterns

##### Stage: 4 and 5 Challenge Level:

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.

### Cubics

##### Stage: 4 and 5 Challenge Level:

Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern.

### Interpolating Polynomials

##### Stage: 5 Challenge Level:

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.

### Motion Sensor

##### Stage: 4 Challenge Level:

Looking at the graph - when was the person moving fastest? Slowest?

### Parabolic Patterns

##### Stage: 4 and 5 Challenge Level:

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.

##### Stage: 4 Challenge Level:

Can you adjust the curve so the bead drops with near constant vertical velocity?

##### Stage: 4 Challenge Level:

Here are some more quadratic functions to explore. How are their graphs related?

##### Stage: 4 Challenge Level:

Explore the two quadratic functions and find out how their graphs are related.

### Climbing

##### Stage: 5 Challenge Level:

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

##### Stage: 4 Challenge Level:

Explore the relationship between quadratic functions and their graphs.

### Cubic Spin

##### Stage: 5 Challenge Level:

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

##### Stage: 4 Challenge Level:

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

### Steve's Mapping

##### Stage: 5 Challenge Level:

Steve has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

### Alison's Mapping

##### Stage: 4 Challenge Level:

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

### Which Is Cheaper?

##### Stage: 4 Challenge Level:

When I park my car in Mathstown, there are two car parks to choose from. Which car park should I use?

### Matchless

##### Stage: 4 Challenge Level:

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

### Real(ly) Numbers

##### Stage: 5 Challenge Level:

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### After Thought

##### Stage: 5 Challenge Level:

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

### Parabella

##### Stage: 5 Challenge Level:

This is a beautiful result involving a parabola and parallels.

##### Stage: 4 Challenge Level:

Four vehicles travelled on a road. What can you deduce from the times that they met?

### Ellipses

##### Stage: 4 and 5 Challenge Level:

Here is a pattern for you to experiment with using graph drawing software. Find the equations of the graphs in the pattern.

### Power Up

##### Stage: 5 Challenge Level:

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

### Graphic Biology

##### Stage: 5 Challenge Level:

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

### Sangaku

##### Stage: 5 Challenge Level:

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

### Graphical Interpretation

##### Stage: 4 Challenge Level:

This set of resources for teachers offers interactive environments to support work on graphical interpretation at Key Stage 4.

### Lap Times

##### Stage: 4 Challenge Level:

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .

### Gosh Cosh

##### Stage: 5 Challenge Level:

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

### Perpendicular Lines

##### Stage: 4 Challenge Level:

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

### Which Is Bigger?

##### Stage: 4 Challenge Level:

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

### Small Steps

##### Stage: 5 Challenge Level:

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

### Curve Fitter

##### Stage: 5 Challenge Level:

Can you fit a cubic equation to this graph?

### Exponential Trend

##### Stage: 5 Challenge Level:

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

### Curve Match

##### Stage: 5 Challenge Level:

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

### Golden Construction

##### Stage: 5 Challenge Level:

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

### Electric Kettle

##### Stage: 4 Challenge Level:

Explore the relationship between resistance and temperature

### What's That Graph?

##### Stage: 4 Challenge Level:

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

### Bus Stop

##### Stage: 4 Challenge Level:

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

### Real-life Equations

##### Stage: 5 Challenge Level:

Here are several equations from real life. Can you work out which measurements are possible from each equation?

### Bio Graphs

##### Stage: 4 Challenge Level:

What biological growth processes can you fit to these graphs?

### Surprising Transformations

##### Stage: 4 Challenge Level:

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

### Equation Matcher

##### Stage: 5 Challenge Level:

Can you match these equations to these graphs?

### Immersion

##### Stage: 4 Challenge Level:

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

### Maths Filler 2

##### Stage: 4 Challenge Level:

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

### Guess the Function

##### Stage: 5 Challenge Level:

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