By recognising the shape of a graph, it is possible to fi nd the rule or mathematical model that describes it. Throughout this chapter, several types of graphs have been investigated.
EXAM TIP Be careful to answer all aspects of the question (this question requires the x-value and the andand maximum). Always re-read the question to check you have met all requirements before moving on to the next question.
[Assessment report 2 2007]
2J
The parabola: y= x2 y
0 x
The graph of a cubic function: y= x3 y
0 x
The hyperbola: y
=1x
x = 0 y = 0
y
0 x
The truncus: y
= x1
2
x = 0 y = 0
y
0 x
The graph of a square root function:
y x
y x
y x
y x
y
0 x
Refl ections and translations can be applied to each of these graphs, but the basic shape of each graph remains the same.
WORKED EXAMPLE 32
Match each of the following graphs with the appropriate model.
i y = ax2 ii y = ax3 iii y a
= x iv y a
= x2 v y a xy ay a a
x
y b
x
y c
x
y d
x
y e
x y
THINK WRITE
Match the graphs using the information in the
summary above. i
is a parabola; it matches graph b. ii is a cubic; it matches graph e.
iii is a hyperbola (the graph is in opposite quadrants);
it matches graph c.
iv is a truncus (the graph is in adjacent quadrants); it matches graph a.
v is a square root function; it matches graph d.
WORKED EXAMPLE 33
x 0 1 2 3 4 5
y 0 2.5 3.54 4.33 5 5.59
The data in the above table exactly fi t one of these rules: y = ax y = ax y =a x
2, y =y = axy =y = axaxax3, ,y = a
x2or y a xy ay ay ay ay ay ay ay a . a Plot the values of y against x. b Select the appropriate rule and state the value of a.
THINK WRITE/DRAW
a Plot the values of y against x. a y
0 x 5 4 3 2 1
1 2 3 4 5 b 1 Study the graph. It appears to be a square
root curve. Write the appropriate rule.
b Assume that y a xy ay a .
2 To fi nd the value of a: select any pair of corresponding values of x and xx y. (Since we need to take a square root, the best to choose is the one where x is a perfect square.)xx
Using (1, 2.5):
3 Substitute selected values into the rule and solve for a.
2 5 1
2 5 2 5= a
= a × 1 a= 2.5
4 We need to make sure that the selected rule is the right one. Replace a with 2.5 in the rule.
Verifying:
y x
y x
y x
y 2 5 x
y x
y x
y x
y x
y x
y x
5 Substitute the values of x from the table xx into the formula and check if you will obtain the correct values of y.
(0, 0): y= 2 5 02 52 5
= 0 (2, 3.54): y = 2 5 22 52 5
= 3.54 (3, 4.33): y = 2 5 32 52 5
= 4.33 (4, 5): y= 2 5 42 52 5
= 5 (5, 5.59): y = 2 5 52 52 5
= 5.59
6 As the values of y obtained by using the rule match those in the table, the choice of model is correct.
The rule that fi ts the data is y a xy ay a , where a= 2.5.
The process of fi tting a straight line to a set of points is often referred to as regression. Statistical data is easiest to deal with in linear form. If the data is not linear, then a linear relationship can still be found by transforming the x scale. A regression line can then be fi tted.xx
For example, y m= += +mmx c
xx is a hyperbola. However, if we substitute X for XX 1
x, the rule becomes linear: y= mX + c. The graph of y versus X will be a straight line with a gradient of XX m and a y-intercept of c. These values (m and c) can then be established from the graph and thus the hyperbolic model can be determined.
Note: In a quadratic relationship, X is substituted for XX x2; in a cubic relationship, X is substituted XX for x3.
WORKED EXAMPLE 34
It is believed that, for the data in the table below, the relationship between x and y can be modelled by y axy ay ay ay axy ay ay ay a= xxxxxxxxxxxxx2++++bx cbbbbx cbbbbbbbbbbx cx cx cx cx cx cx c++++ .
x 0 1 2 3 4 5
y 4 5.3 8.6 14.8 23 34.4
a Plot the values of y against x. b Calculate the values of a, b and c (correct to 3 decimal places) and write the equations.
THINK WRITE/DISPLAY
a 1
2
On a Lists & Spreadsheet page, enter x-values into column A and y-values into column B. Label each column x and y respectively.
On a Data & Statistics page, move the pointer to the horizontal and vertical axes and select the x-values variable and y-values variable respectively.
b 1
2 Interpret the variables given on the screen.
a= 1.252 b=−0.222 c= 4.096
∴ y = 1.252x 1.252 1.252 2− 0.222x 0.222 0.222 + 4.096 Correct to 3 decimal places.
If the relationship between the variables is not given, we have to make an assumption of a model from the graph of the data. We then have to transform the data according to our assumption.
If the assumption was correct, the transformed data, when plotted, will produce a perfectly straight, or nearly straight, line.
Note: In this section we will consider only the rules of the type y= ax2+ b, y = ax3+ b,
y a
x b
= +
= + and so on (we will not allow for a horizontal translation), so that the appropriate substitution can be made.
WORKED EXAMPLE 35
x 1 2 3 4 5 6
y 35 21 16 12 11 10
Establish the rule connecting x and y that fi ts these data.
THINK WRITE/DRAW
1 Using either graph paper or a CAS
calculator, plot y against x. y
0 x 3530 2520 1510 5
1 2 3 4 5 6
2 The scatterplot appears to be a hyperbola. Write the appropriate formula (remember that we do not consider horizontal translations in this section).
Assumption: y a
= +x b
= + Return to the Lists & Spreadsheet
page, then press:
• MENU b
• 4:Statistics 4
• 1:Stat Calculations 1
• 6:Quadratic Regression 6 Change the X List to x, and the Y List to y.
Select OK and press ENTER ··
3 Check your assumption: prepare a new table by replacing values of x with xx 1 (leave the values of y unchanged). x
1
x 1 0.5 0.33 0.25 0.2 0.17
y 35 21 16 12 11 10
4 Plot y against 1
x. y
0 x 3530 2520 1510 5
0.2 0.4 0.6 0.8 1 1–
5 Comment on the shape of the graph. The graph is very close to a straight line,
therefore the assumption of a hyperbolic model is correct.
6 If we replace 1
x with X, the rule becomes y= aX + b, which is the equation of the straight line, where a is the gradient and b is the y-intercept.
These (a and b) can be found from the graph as follows: draw in the line of best fi t.
y
0 3530 2520 1510 5
0.2 0.4 0.6 0.8 1 X 7 Write the formula for the gradient. m y y
x x
= yy yy
x x
x x
2 1
y y
y y
2 1
x x
x x
8 Select any 2 points on the line. Using (0.17, 10) and (1, 35):
9 Substitute the coordinates of the points into the formula and evaluate.
m= 35 10− 1 0−− 17.
= 25 0 83 0 8
= 30.120 8
10 Write the value of a. Since a is the gradient, a= m
= 30.12.
11 Write the general equation of the straight line.
y= mx + c
12 Substitute the value of m and the coordinates of any of the 2 points, say (1, 35) into the equation.
35 = 30.12 × 1 + c
13 Solve for c. (Alternatively, read the
y-intercept directly from the graph.) 35 = 30.12 + c c= 35 − 30.12
= 4.88
14 State the value of b. Since b is the y-intercept, b cb cb c 4 88= =4 84 888. .
15 Substitute the values of a and b into
y a
= +x b
= + to obtain the rule that fi ts the given data.
The rule for the given data is:
y==30 12x ++ . 4 88
4 8 4 8
Modelling
1. is the process of fi nding the rule that fi ts the given data.
The rule itself is called a
2. mathematical model.
The best way to start modelling is to produce a scatterplot of the original data.
3.
Use the scatterplot of the data to make an assumption of the model of the relationship.
4.
It should be of the type y= ax2+ b, y = ax3+ b, y a x b
= +
= + and so on. To test the assumption, transform the data accordingly. If the assumption is correct, the transformed data when plotted will produce a straight, or nearly straight, line.
To fi nd the values of
5. a and b in the model, draw a line of best fi t; a is the gradient of the line and b is the y-intercept.
REMEMBER
Modelling
1 WE 32 Match each of the graphs with the appropriate model:
i y = ax2+ b ii y = ax3+ b graph. Select the most appropriate rule, and fi nd the value of a.
a x −3 −2 −1 0 1 2 3 eBookplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus eBoo eBookkkplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus
Digital doc Spreadsheet 076 Modelling
c x −5 −2 −1 1 2 5
y 0.08 0.5 2 2 0.5 0.08
d x 0 0.5 1 1.5 2
y 0 1.13 1.6 1.96 2.26
e x 1 2 4 5 10
y 5 2.5 1.25 1 0.5
f x −3 −2 −1 0 1 2
y 40.5 12 1.5 0 −1.5 −12
3 MC Which of the graphs below could be modelled by y a
x b
= +
= 2 +
= +
= + ?
i y
x
ii y
x
iii y
x
iv y
x
v y
x
A i only B i,ii and iii C iv and v
D i,ii and iv E i,iv, and v
4 WE 34 It is believed that for the data in the table below, the relationship between x and xx y can be modelled by y = ax2+ b.
x 0 1 2 3 4 5
y −3.2 −1 4.9 14.5 29 46.8
a Plot the values of y against x.
b Plot the values of y against x2 and draw the line of best fi t.
c Find the values of a and b and hence the equation describing the original data.
5 The table below shows the values of 2 variables, x and xx y.
x −4 −2 0 2 4 6
y −28 −13.5 −12.5 −10 4.3 41
Establish the mathematical model of the relationship between the variables, if it is known that it is of the form y = ax3+ b.
6 The table below shows the results, obtained from an experiment, investigating the frequency of a sound, f, and the length of the sound wave, ff λ.
λ 0.3 0.5 1 3 5 8 10
f 1130 680 340 110 70 40 35
a Plot f against ff λ.
b From the following relationships select the one which you think is suggested by the plot:
f= aλλλ , f2 a
=λ,
λλ fff aaa λ .
c Based on your choice in part b, plot f against either ff λλλ , 2 1
λ or λ , draw in the line of best fi t and use it to fi nd the rule that connects the 2 variables.
7 For her science assignment, Rachel had to find the relationship between the intensity of the light, I, and the distance between the observer and the source of light, II d. From the experiments she obtained the following results.
d 1 1.5 2 2.5 3 3.5 4
I 270 120 68 43 30 22 17
a Use a graphics calculator to plot the values of I against II d. What form of relationship does the graph suggest?
b Nathan (Rachel’s older brother) is a physics student. He tells Rachel that from his studies he is certain that the relationship is of the type I a
=d2. Use this information to help Rachel to fi nd the model for the required relationship.
8 WE 35 The table below gives the values of 2 variables, x and xx y. Establish the rule, connecting x and
xx y, that fits these data.
x 0 1 3 5 7 9
y 4 7 9 11 12 13
9 Joseph is a financial adviser. He is studying the prices of shares of a particular company over the last 10 months.
Months 1 2 3 4 5 6 7 8 9 10
Price, $ 6.00 6.80 7.45 8.00 8.50 8.90 9.30 9.65 10.00 10.30 a Represent the information graphically.
b Establish a suitable mathematical model, which relates the share price, P, and the number of the month, m.
c Use your model to help Joseph predict the share price for the next 2 months.
eBoo eBookplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus eBoo eBookkkplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus
Goal accuracy
SUMMARY
Graphs of the power functions
Name Equation Basic shape Domain Range Special feature
Parabola y= a(x − b)2+ c yy
0 x ((b, c)
R If a> 0
y≥ c If a< 0 y≤ c
Turning point at (b, c)
Cubic y= a(x − b)3+ c yy
0 (b, cb, c)x
R R Stationary point of
infl ection at (b, c)
Hyperbola y a
x b c
= x bx b+ or y a x b= − c y a
y a== (x bx bx bx bx bx b−− )−1+ yy
x b c
0
R \{b} R \{c} Horizontal asymptote y= c, vertical asymptote x= b
Truncus y a
x b c
= +
(x bx bx bx bx bx b)2 or y a x b= − c y a
y a== (x bx bx bx bx bx b−− )−2+ yy
b x c
0
R \{b} If a> 0 y> c If a< 0 y< c
Horizontal asymptote y= c, vertical asymptote x= b
Square root y a x b cy ay a x b== x b−− + or y a x b= − c y a
y a== (x bx bx bx bx bx b−− )12+ yy
0 ((b, c) x
x≥ b If a> 0 y≥ c If a< 0 y≤ c
End point at (b, c)
The equation for any graph
• y= f (x) above can be written in the general form:
y= af (x − b) + c.
This form can be used to describe transformations of all of the functions considered.
For all of the above functions:
• a
1. is the dilation factor: it dilates the graph from the x-axis.
When an equation for these types of graphs is put into its general form of
2. y= af (x − b) + c, the horizontal
dilation can be described in terms of a vertical dilation.
If
3. a< 0, the basic graph is refl ected in the x-axis.
f
4. (b− x) or f (−x+ b) is the refl ection of f (x + b) in the y-axis.
b
5. translates the graph b units along the x-axis (to the right if b> 0, or to the left if b < 0).
c
6. translates the graph c units along the y-axis (up if c> 0, or down if c < 0).
To put equations into general form:
•
If the coeffi cient of
7. x is a number other than 1, to fi nd the value of xx b and a, the equation should be transposed to make the coeffi cient of x equal to 1.xx
For example, y = (3x + 5)2+ 4 The absolute value function y
Sketch the graph of 1. y = f ( f f x).
Refl ect the portion of the graph that is below the
2. x-axis in the x-axis.
Or:
Express the function in hybrid form with specifi c domains where the absolute value expression is positive 1.
and negative.
Sketch each rule for the specifi ed domain.
2.
For functions of the form
• y= a| f| | (ff x)| + c, a and c have the same impact on the graph of the absolute value function, as on the graphs of all other functions discussed in this section.
Transformations with matrices
The use of matrices to map transformations of points and equations can be summarised as follows, where
• (x′, y′) is the image of the point (x, y) under the transformation.
T x
y represents a refl ection in the y-axis.
T x
y represents a refl ection in the x-axis.
T x
y represents a dilation of a factor of a from the y-axis.
T x
ay represents a dilation of a factor of a from the x-axis.
Transformations can be combined to represent more than one transformation. For example,
•
describes the following: dilation by a factor of 4 from the y-axis, a dilation by a factor of 12 from the x-axis, refl ection in the x-axis, a horizontal translation of +2 and a vertical translation of +3.
Sum and difference of functions For the graph of the
• sum/difference function, dom (f function, dom ( function, dom ( (x) ± g(x)) = dom f (x) ∩ dom g(x). The graph of the sum/difference function can be obtained by using the addition of ordinates method.
For the
• product function, dom (productproduct function, dom ( function, dom ( (x)g(x)) f = dom f (x) ∩ dom g(x). Some features of the graph of the product function are as follows:
° the x-intercepts of f (x)g(x) occur where either f (x) or g(x) have their x-intercepts
° f (x)g(x) is above the x-axis where f (x) and g(x) are either both positive or both negative
° f (x)g(x) is below the x-axis where one of the functions f (x) or g(x) is positive and the other is negative.
Composite functions and functional equations For the
• composite function f (g(x)) to be defi ned, the range of g must be a subset of the domain of f. ff Furthermore, if f (g(x)) is defi ned, the domain of f (g(x)) equals the domain of g(x).
Equations involving algebra of functions, for example
• f (2x(2(2 ) = 2f 22 (x), are generally tested to determine if they are true for particular functions.
° To determine if an equation is true for a particular function, consider the LHS and RHS of the equation separately to determine if the equation holds true for all values of x.
° Alternatively, you may fi nd a particular x-value for which the equation does not work; that is, a counterexample.
° These types of equations can be investigated by defi ning the functions on the CAS calculator and then testing the algebraic function equation.
Modelling
Modelling is the process of fi nding the rule (mathematical model) that fi ts the given data.
•
To model:
•
Plot the original data on graph paper or use a CAS calculator.
1.
Make an assumption of the model.
2.
Transform the data in accordance with your assumption.
3.
Check the assumption by plotting the transformed data (if correct, the graph will be a straight or nearly 4.
straight line).
Draw in a line of best fi t.
5.
Find the equation of the line (
6. y = mx + c).
Replace
7. x in the equation with the transformed variable (for example, xx x2,1 x).