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# Grade 9 Ext EOY 2017 - PartB

- Author:
- Callum Marshall

## Question 1

The graph below shows the equation . Set the value of the slider so that b = 0.
i. EXPLAIN why, when b = 0 the vertex of the parabola is at (0, -2).

Turn on the TRACE function for the vertex by checking the box shown and slowly change the value "b" using the slider.
ii. Describe the shape of the line traced out by the vertex as the value of parameter b is changed.

## Question 2

The graph below shows the equation . The point A (1, -5) has now been marked on the diagram.

Turn the TRACE on for the vertex and as before, change the value of the parameter b.
i. Find the equation of the trace line formed by the vertex as the value of parameter b is changed.
ii. Comment on any relationship you notice between the parameter values of the original equation and those for the equation of the trace line.

## Question 3

The graph below shows the equation .
i. Find the equation of the line traced out by the vertex as parameter b is changed, explaining your method clearly.
ii. Use the input bar to verify your equation.
iii. Again, comment on any relationship between the parameter values of the original equation and those you found for the equation of the trace line.

## Question 4

The graph below shows the general quadratic equation
i. Using the sliders to set the values for parameters a and c, explore further the relationship you have described previously.

**Use the table to help organise your data****.**ii. What conclusions can you come to based on your results so far?
iii. Where is the general quadratic function and is the trace function, describe the relationship between the fixed parameter values of the original quadratic function and the trace equation of the line generated as the value of parameter b is changed.

## Question 5

In this question you will try to justify why the relationship you found makes sense.
i. Using , to represent the general quadratic function and to represent the trace function EXPLAIN why .
ii. Hence, by substituting into each function, simplifying each side of the equation and then comparing them, justify the relationship you described in Question 4.
.

**Hint for part (ii):**use p, q, ... for the parameters for the trace function## Question 6

The graph below shows the equation .
The point A is marked on the parabola. When b = 0 the coordinates of the point A are (1, 1).

i. The equation of the trace line for the point A, as the value of the parameter b is changed, can be written in the form . Find the values for p, q and r.
ii. Using your result from Question 1 describe the relationship between the equation for the trace line of the vertex of the parabola and the equation of the trace line you found in part (i).

## Question 7

The graph below shows the equation .
The point A is marked on the parabola. When b = 0 the coordinates of the point A are

i. As before, the equation of the trace line of point A can be written in the form . Find values for p, q and r.
ii. Describe the relationship between the equation for the trace line of the vertex of the parabola and the equation of the trace line you found in part (i).

## Question 8

The graph below shows the equation .
Two points A and B have been marked on the graph. When , A is at and B is at (-2, -5).

**Use the diagram to further explore the relationship(s) you described in the previous two questions.**## Question 9

A parabola has equation , where parameter b varies as seen above.
i. Write down the equation of the trace line for the vertex.
When the point A (2, 7) lies on the parabola.
ii. Write down the equation of the trace line for point A.
iii. EXPLAIN how you used the relationships you described in the previous questions (6, 7 and 8) to get your result for part ii) above.

## Question 10

A parabola has equation , where parameter b can vary (as above).
i. Write down the equation of the trace line for the vertex.
When the point A (p, q) lies on the parabola.
ii. Write down the equation of the trace line for point A.
iii. JUSTIFY your answer to part ii) above.