non-linear relationships Flashcards
how do exponential functions work?
- 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
how is an exponential function graphed?
- construct a table of values
- draw a number plane
- plot the points
- join the points to make a curve
what are the y values for an exponential function y = 3^x?
- 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
what are the y values for an exponential function y = 3^-x?
- 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
how does an exponential model work?
- 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
what is a quadratic function?
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
how is a parabola graphed?
- construct a table of values
- draw a number plane
- plot the points
- join the points to make a parabola
what are the y values for a quadratic function y = x² + 1?
- 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
what is a quadratic model?
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
how is stopping distance calculated?
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
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
d = 5vt + v²
—– —–
18 170
= 5 x 60 x 0.8 + 60²
—————- —–
18 170
= 34.50980…
≈ 35m
what is a reciprocal function?
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
how is a hyperbola graphed?
- construct a table of values
- draw a number plane
- plot the points
- join the points to make a hyperbola
what are the y values for a reciprocal function y = 2/x?
- 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
what is a reciprocal model?
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
what is an inverse variation?
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
how is an inverse variation problem solved?
- 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
- solve the equation for k by substituting values for x and y
- write the equation with the solution for k (step 2) and solve the problem by substituting a value for either x or y
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?
y = k
—
x
c = k
—
n
36 = k
—
50
k = 1800
c = 1800
——-
n
= 1800
——–
20
= $90
∴ cost is $90 per person
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?
y = k
—
x
c = k
—
n
36 = k
—
50
k = 1800
c = 1800
——-
n
25 = 1800
——-
n
n = 72
∴ 72 people required
what is algebraic modelling?
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
