SHORT ANSWER
1 For the function x − 3)2− 4:
a state the coordinates of the turning point b state the domain and range
c sketch the graph.
2 The graph of y a
x b c
= x bx b+ has a vertical
asymptote at x= 2 and a horizontal asymptote at y=−1.
a Find the values of b and c.
This graph then undergoes the following transformations:
refl ection in the
• x-axis
dilation by a factor of 3 from the
• x-axis
horizontal shift of 2 units right.
•
b If the intersection of the two graphs is at (m, 2), fi nd the value of m.
c Hence fi nd the equation of the transformed graph.
x c. Hence describe the transformations required to produce this curve from the graph of
y=1x .
4 The graph of a cubic function has a stationary point of inflection at (1, 1). It cuts the y-axis at y = 4.
Find the equation of the graph.
5 The graph of y
=1 was dilated by the factor ofx 4 from the x-axis, refl ected in the x-axis and then translated 2 units to the left and 1 unit down.
a State the equation of the asymptotes.
b State the domain and range.
c State the equation of the new graph.
d Sketch the graph.
6 a State the changes necessary to transform the graph of y
= x1
2 into the one shown.xx b Find the equation of
the graph.
7 The domain of a truncus is R \{1}, the range is (−∞, 2) and the graph cuts the y-axis at y =−3. Find the equation of the function.
8 The basic square root curve was reflected in both axes and then translated so that its intercepts at the axes were (0, 1) and (−5, 0). Find the size and the direction of the translations; hence, find the equation of the new graph.
9 a Sketch the graph of y= −= −= −2 2 −
2 2 (xx++ 22) , clearly showing the coordinates of the cusps, the intercepts with the axes and the position of the asymptotes.
b State the domain and range of the graph in a.
10 The graph of f: [ff −5, 1] → R where
b State the range of the function with the rule y= /f // (x)/)) and domain [// −5, 1].
[© VCAA 2006]
11 The point (−1, 3) undergoes a translation given by the matrix a
12 The point (1, 2) undergoes a series of transformations given by the matrices a
b
b Find the image under the transformations of:
i yyyyyy 2 xxxxxx ii y= x3+ x
13 A point on a curve (xA point on a curve (A point on a curve ( , y) undergoes a transformation descibed by a 0
0 2 where a is a real constant such that a > 0.
y
−2 x
−1
–2
− 223
a Find the values of x and xx y in terms of a, x' and y'.
b If the point is on the curve y= 2x 2 2 2− x, fi nd the image of the curve in terms of a under this transformation. set of axes. Hence, sketch (f
set of axes. Hence, sketch (
set of axes. Hence, sketch ( + g)(x). function on the same set of axes. Hence, sketch f (g(x)). scatterplot to choose a suitable model.
b Plot the values of y against either x3, 1 x2 or x (depending on your choice in part a). Did you choose the right model? Explain your answer.
c Find the value of a.
MULTIPLE CHOICE
1 The equation of a parabola is given by
y = m − 2(x 2(2( + 3)2, where m> 0. The increase in m will result in:
A the graph being thinner B the graph being wider C the increase of the domain D the increase of the range
E the graph being shifted further to the right 2 The coordinates of the turning point of the parabola
y = 2(3x + 6)2− 3 are:
A (6, −3) B (−6, −3) C (2, −3) D (−2, 3) E (−2, −3)
3 The graph of yyyyyyyyyyyyyyyyyyy==23(bbbbbbbbbbbbbbbxxbbbbxxxx 3−−33)333++111 is dilated in the y direction by the factor of:
A 23 B 23b C 2
A the horizontal asymptote y = 2 B the horizontal asymptote y = 1 C the horizontal asymptote y = 3 D the vertical asymptote x = 2 resulting graph would have the equation:
A y
C y
A The vertical asymptote is x = 2.
y-intercept is greater than −4.
9 To obtain the graph shown, we need to:
A translate the graph of y= x1
2 one unit to the left and refl ect in the x-axis B translate the graph of
y= x1
translate the graph of translate the graph of
2 one unit to the left and refl ect in the y-axis C translate the graph of y
= x1
2 one unit to the left and refl ect in both axes
D translate the graph of y
= x1
2 one unit to the left and dilate it in the x directionxx
E none of the above
10 The equation of this graph is of the form:
A y a x m ny ay a x m== x m−− + , a > 0 B y a m x ny ay a m x== m x−− + , a > 0 C y a x m ny ay a x m== x m−− + , a < 0 D y a m x ny ay a m x== m x−− + , a < 0 E y a x m ny ay a x m== x m++ + , a < 0 11 The equation of this graph
could be:
13 The range of the function y
= − x
14 The equation of the graph shown in the diagram below is best described by:
A y = |x + 2| + 2
19 The data in the following table exactly fit one of
20 For certain data the values of y were plotted against 1
xand the line of best fit was drawn as seen on the diagram below. The model that relates the
a Sketch the graph of each of the following functions on the same set of axes with the original graph and give the coordinates of the points A, B, C and D.
i y=−−−f (x) ii y= f (−x) iii y= f (x − 2)xx iv y= f (x) + 3
v y= 2f 2 2 (x) vi y= 1 − f (x + 1)
b Maya, a fabric designer, wishes to use the curve of y= f (x) (red) to create a ‘wavy’ pattern as shown in the diagram at right.
If she wants the waves to be 2 units apart vertically, suggest the best way she could alter the equation of y= f (x). (Remember a fabric has a fi xed width!)
2 Consider the function f:R→ R, f(ffx) = (x − 1)2 (x − 2) + 1.
a The coordinates of the turning points of the graph of y= f(ffx) are (a, 1) and (b,2527). Find the values of a and b.
b Find the real values of p for which the equation f(ff x) = p has exactly one solution.
c For the following, k is a positive real number.kk
i Describe a sequence of transformations that maps the graph of y= f (x) onto the graph of y f x
EXAM TIP Students must ensure that they show their working — if a question is worth more than one mark, students risk losing all available marks if only the answer is given and it is incorrect. The
instructions at the beginning of the paper state that if more than one mark is available for a question, appropriate working must be shown.
When students present working and develop their solutions, they should use conventional mathematical expressions, symbols, notation and terminology.
[Assessment reports 1 and 2 2007 © VCAA]
The graph of y= f (x) is to be transformed to become the graph of y = f (2x(2(2 ) + 1.
a Describe these two transformations.
b Create matrices to represent these transformations.
c Use these matrices to fi nd the images of the points (1, 1) and (2, 0.25) under these transformations, and use these values to deduce the images of the points (−1, 1) and (−2, 0.25).
d On the same axes, sketch this transformed function, showing the coordinates of the four points from c above.
e Using any method, fi nd a rule for this transformed function.
4 A proposed section of a ride at an amusement park is to be modelled on the curve yyyyyyyyyyyyyyyyyyyyyyyyyyyy========= 5001 (600600600xxxxxxxxxxxxxxxxxxxxxxxxxxxx 25+++++++++2525xxxxxxxxxx222 xxxxxxxxxx ,333) where y is the height (in metres) of the ride above ground level and x is the horizontal distance (in metres). xx The x-axis represents ground level. It will travel through a tunnel from A to C; B is the lowest point in the tunnel and D is the highest point on the ride.
y
A x B
C E
D
a Find the horizontal distance from A to E.
b Find the greatest depth below ground level and the maximum height above ground level that the rollercoaster will reach in this section (correct to 2 decimal places).
c Describe the impact that a dilation by a factor greater than 1 from the x-axis would have on:
i the maximum depth and maximum height from b
ii the point at which the rollercoaster would emerge from the tunnel iii the gradient of the slope at this point.
5 Lena and Alex are planning to buy a new house. They’ve been watching the prices of 3-bedroom houses in a specific area, where they want to live, for the whole year. During each month they collected the data and then, at the end of the month, they calculated the average price for that month. The results of their calculations are shown in the table below. (The prices given are in thousands of dollars.)
Month 1 2 3 4 5 6 7 8 9 10 11 12
Price 240 248 255 261 266 271 273 274 275 274 272 270 a Plot the prices against the months. What model does the graph
suggest?
b If the model of the form y= a(x − b)2+ c is to be used for these data, what is (judging from the graph) the most suitable value for h?
c Plot the values of y (the prices) against (x− b)2, where b is the value you’ve selected in part b. Comment on the shape of the graph.
d Draw a line of best fi t and fi nd its equation. Hence, state the values of a and c in the model.
e Write the equation of the model.
f According to the Real Estate Institute, the property market is on a steady rise (that is, the prices are going up and are likely to rise further). Do the data collected by Lena and Alex support this theory?
g Use the model to predict the average price for the next 2 months.
h Lena and Alex were planning to spend no more than 250 000 for their new house. Several months ago the prices were in their range, but they could not fi nd what they wanted. If the prices are going to behave according to our model, how long do they have to wait until the prices fall back into their range?
6 An eagle soars from the top of a cliff that is 48.4 metres above the ground and then descends towards
unsuspecting prey below. The eagle’s height, h metres above the ground, at time t seconds can be modelled by tt
the equation h a
= +t
b Find the eagle’s height above the ground after i 5 seconds ii 20 seconds.
c After how many seconds will the eagle reach the ground?
d Comment on the changes in speed during the eagle’s descent.
e Sketch the graph of the equation.
After 24 seconds, the eagle becomes distracted by another bird and reaches the ground exactly 2 seconds later. For this second part of the journey, the relationship between h and t can be modelled by the equation tt h= a(t − 24)2+ c.
f Find the values of a and c.
g Fully defi ne the hybrid function that describes the descent of the eagle from the top of the cliff to the ground below. eBookplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus eBoo eBookkkplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus
Digital doc Test Yourself Chapter 2