non-linear relationships

Question 1

how do exponential functions work?

Answer 1

• exponential functions have x as the power of a constant (e.g. 3^x). they are defined by the general rule y = a^x and y = a^−x where the constant a > 0
• most practical uses of exponential functions have a > 1 (k is constant); when a > 1, the graph goes up/down to the right
• when a is greater than 0 but less than 1, the shape of the curve is reversed horizontally (k is constant); when a < 1, the graph goes up/down to the left
• when a = 1 the graph is flat line, y = k

Question 2

how is an exponential function graphed?

Answer 2

1. construct a table of values
2. draw a number plane
3. plot the points
4. join the points to make a curve

Question 3

what are the y values for an exponential function y = 3^x?

Answer 3

• x = -3 - y = 1/27
• x = -2 - y = 1/9
• x = -1 - y = 1/3
• x = 0 - y = 1
• x = 1 - y = 3
• x = 2 - y = 9
• x = 3 - y = 27

Question 4

what are the y values for an exponential function y = 3^-x?

Answer 4

• x = -3 - y = 27
• x = -2 - y = 9
• x = -1 - y = 3
• x = 0 - y = 1
• x = 1 - y = 1/3
• x = 2 - y = 1/9
• x = 3 - y = 1/27

Question 5

how does an exponential model work?

Answer 5

• exponential growth - quantity increases rapidly according to the function y = a^x where a > 1
• exponential decay - quantity decreases rapidly according to the function y = a^-x where a > 1

Question 6

what is a quadratic function?

Answer 6

a quadratic function is a curve whose equation has an x squared (x²). it is defined by the general rule
y = ax² + bx + c where a, b and c are numbers. quadratic functions are graphed in a similar method to exponential functions except the points are joined to make a curve in the shape of a parabola

Question 7

how is a parabola graphed?

Answer 7

1. construct a table of values
2. draw a number plane
3. plot the points
4. join the points to make a parabola

Question 8

what are the y values for a quadratic function y = x² + 1?

Answer 8

• x = -3 - y = 10
• x = -2 - y = 5
• x = -1 - y = 2
• x = 0 - y = 1
• x = 1 - y = 2
• x = 2 - y = 5
• x = 3 - y = 10

Question 9

what is a quadratic model?

Answer 9

a quadratic model describes a practical situation using a function in the form y = ax² + bx + c where a, b and c are numbers. quadratic functions are graphed to make a curve in the shape of a parabola

Question 10

how is stopping distance calculated?

Answer 10

stopping distance = reaction distance + braking distance

d = 5vt + v²
—– —–
18 170

where d - stopping distance in metres
v - velocity or speed of the motor vehicle in km/h
t - time reaction in seconds

Question 11

Tahlia was driving at a speed of 60 km/h and reaction time of 0.80 seconds. calculate the stopping distance correct to the nearest metre

Answer 11

d = 5vt + v²
—– —–
18 170

= 5 x 60 x 0.8 + 60²
—————- —–
18 170
= 34.50980…
≈ 35m

Question 12

what is a reciprocal function?

Answer 12

a reciprocal function is a curve whose equation has a variable in the denominator such as 1/x. it is defined by the general rule y = k/x where k is a number. reciprocal functions are graphed in a similar method to other non-linear functions and make a curve called a hyperbola

Question 13

how is a hyperbola graphed?

Answer 13

1. construct a table of values
2. draw a number plane
3. plot the points
4. join the points to make a hyperbola

Question 14

what are the y values for a reciprocal function y = 2/x?

Answer 14

• x = -4 - y = -0.5
• x = -2 - y = -1
• x = -1 - y = -2
• x = -0.5 - y = -4
• x = 0.5 - y = 4
• x = 1 - y = 2
• x = 2 - y = 1
• x = 4 - y = 0.5

Question 15

what is a reciprocal model?

Answer 15

a reciprocal model describes a practical situation using a function in the form y = k/x where k is a number. reciprocal functions are graphed to make a curve in the shape of a hyperbola

Question 16

what is an inverse variation?

Answer 16

inverse variation (or inverse proportion) occurs when one variable increases while the other variable decreases. the variables are dependent on each other but change in opposite directions. for example, the time taken to paint a house depends inversely on the number of people available to paint; the more painters available the less time it would take to paint the house

Question 17

how is an inverse variation problem solved?

Answer 17

1. write an equation relating the two variables (k is the constant of variation). y is inversely proportional to x so the equation is y = k/x
2. solve the equation for k by substituting values for x and y
3. write the equation with the solution for k (step 2) and solve the problem by substituting a value for either x or y

Question 18

the cost per person (c) to hire a reception centre is inversely proportional to the number of people attending (n). if there are 50 people, the cost per person is $36. what is the cost per person when there are 20 people attending?

Answer 18

y = k
—
x

c = k
—
n

36 = k
—
50
k = 1800
c = 1800
——-
n
= 1800
——–
20
= $90

∴ cost is $90 per person

Question 19

the cost per person (c) to hire a reception centre is inversely proportional to the number of people attending (n). if there are 50 people, the cost per person is $36. how many people are required for the cost per person to be $25?

Answer 19

y = k
—
x

c = k
—
n

36 = k
—
50
k = 1800
c = 1800
——-
n
25 = 1800
——-
n
n = 72

∴ 72 people required

Question 20

what is algebraic modelling?

Answer 20

algebraic modelling occurs when a practical situation is described mathematically using an algebraic function. this involves gathering data and analysing the data to determine possible functions. determining the function is made easier using technology