Friday, March 11, 2016
Page |1
Exemplar for Part One of the Graphs internal on 9th of April 2014
NB: The actual task will be similar to this. It could also include a square root function.
AS92157 Apply Graphical Methods in Solving Problems.
Achieved:
-
Selecting and using methods
Demonstrating knowledge of properties of functions and graphs
Communicating using appropriate representations
Polly took a photo of her friends screen because she wanted to learn how to
make patterns like this one. She requests that you tell her what functions
create the picture above, and explain how she would be able to work out the
functions herself.
Polly wants you to graph the equations y=-2|x+0.5|+3 and y=-0.5(x+2)(x-1)
because she saw them written on her friends book and wondered if they
would complete the picture.
The clarity and completeness of your answer will determine your grade.
Friday, March 11, 2016
Page |2
Exemplar for Part One of the Graphs internal on 9th of April 2014
Achieved Level Answers:
The red curve has the shape of a parabola y=x^2
Use the x-intercepts to form the equation.
The x intercepts are (-7,0) and (-2,0), use these to form factors of your function.
The x intercept of -7 indicates a factor of (x+7)
The x intercept of -2 indicates a factor of (x+2)
We must calculate the stretch factor K, y=k(x+7)(x+2)
To solve for K pick any point, (-10,6) is an easy one to see clearly on the graph.
6=k(-10+7)(-10+2)
6=-3*-8*k
K=6/24 which we round to 0.25
So the equation of the red graph is y=0.25(x+7)(x+2)
Use values of x between -7 and -2 to form the bit of the graph you require.
The blue graph has the shape of a cubic y=x^3
Use the x intercepts to form the equation.
The x intercepts are at -7, -2 and 1. Change their signs when using these roots to form the equation
in factored form. You also need to add a stretch factor K.
The equation is y=k(x+7)(x+2)(x-1) and you need to substitute in a point in order to calculate k.
I have chosen the point (-3,3) so change y to -2 and change x to 3 to get:
3=k(-3+7)(-3+2)(-3-1)
3=k*4*-1*-4
K=3/16=0.187 round it to 0.2
Friday, March 11, 2016
Page |3
Exemplar for Part One of the Graphs internal on 9th of April 2014
So the equation for the blue curve is y=0.2(x+7)(x+2)(x-1)
Use x values between -7 and 1 to draw the piece of the graph required.
Graphing this equation: y=-2|x+0.5|+3
The base graph is y=|x| where the 3 moves it up 3 units; to 0.5 moves it left 0.5 units and the -2
gives the gradient of the RHS of the point (-0.5,3)
The point at the top of the V is (-0.5,3), the -2 means the V is upside down.
Graphing this equation y=-0.5(x+2)(x-1)
The base function is y=x^2.
The -0.5 tells us the function is upside down.
The (x+2) tells us that when x=-2; y=0 giving the x intercept of (-2,0)
The (x-1) tells us that when x=1;y=0 giving the x intercept of (1,0)
The x coordinate of the vertex will be in the middle of -2 and 1, because parabolas have symmetry.
The x coordinate of the vertex is -0.5, so substitute this into y=-0.5(x+2)(x-1) to get the y value of the
vertex. y=-0.5(-0.5+2)(-0.5-1)=1.125
Plot the three points and join them up with a smooth parabola with the vertex at (-0.5,1.125)
Friday, March 11, 2016
Exemplar for Part One of the Graphs internal on 9th of April 2014
Page |4
Friday, March 11, 2016
Page |5
Exemplar for Part One of the Graphs internal on 9th of April 2014
Excellence: Even an Achieved level student should aim to comment on excellence combination
graphs during the assessment on the 10th of April.
Polly was amazed by these graphs and wants some advice on how to create graphs like these.
Achieved Level Answers:
The blue function is a combination of the absolute value of x and sine.
I can see the V shape of the absolute value function.
I can see the sine wave clearly and can also see it has been compressed in the horizontal direction.
The red function is a combination of an upside down parabola and the sine wave.
I can see the upside down parabola shape clearly with roots at around 7.5 and 18.5.
I can see the shape of the sine wave with a compression in period.
The green function looks like y=x-3 combined with a sine wave.