graphs and functions

Question 1

what is a function?

Answer 1

a mathematical process that uniquely links one or more variables to another variable eg advertising to sales

Question 2

what does a function always have?

Answer 2

an input (domains)

a rule for processing the input(s)

an output

Question 3

what are some real world functions?

Answer 3

business profit calculation: revenue- costs = profit

finance loan interest: principle amount we borrow x interest rate x time

Question 4

what makes a function a function?

Answer 4

for each value of x (input) there is only ONE corresponding value of y (output) there can only be one value of y for a value of x

Question 5

difference between x and y?

Answer 5

x is along the corridor y is up the stairs

Question 6

how would you know if something was a function of something else?

Answer 6

if y = 0.4x + 50z then y is a function of x and z or in short y = f(x, z)

Question 7

generate the function definition and draw the graph showing unit costs for the example.
a compaany charges £2 per unt up to 100 kg with a 20% discount for orders 100kg or more and 30% for 200kg or more

Answer 7

first find how much the discounts would be
100 kg up to = 2
100kg or more = 2 x 0.8 = 1.60
200kg or more = 2 x 0.7 = 1.40

plot these on a graph using constant lines across if all values were the same there would be one long constant line

Question 8

what is the linear function?

Answer 8

y = ax + b

gradient = a
intercept = b

Question 9

in a linear function what happens when y increases or decreases?

Answer 9

it increases by a constant amount of x
eg for y = ax + b then if we increase x by one unit y will increase by the amount a

Question 10

how do you find the gradient?

Answer 10

(change in y) - (change in x)

Question 11

what is the intercept?

Answer 11

point ar which function cuts the vertical axis

Question 12

what is the gradient?

Answer 12

steepness of the slope if we change the x then y will change by the gradient

y =10x + 100 this means that if x changed by 1 y would change by 10 and 100 is the constant

Question 13

when drawing a function what must we consider?

Answer 13

put it into the function of y = ax = b to find y when x = certain points on your graph

Question 14

how to find the equation of the line?

Answer 14

put both brackets into y = ax + c equation
rearange first equation and put both of them togther
then shift everything until you have the value of a
find the value of b

or to find the gradient For example on a straight line with points (4, 2) and (6, 8) we take the difference between the y coordinates (8 – 2 = 6) and the difference between the x coordinates (6 – 4 = 2) , divide 6 by 2 and we have found a gradient of 3 .

Question 15

is a linear function for straight or curved lines?

Answer 15

straight

Question 16

why cant we use a straight line to join the points on a curved graph?

Answer 16

because it would be inaccurate

Question 17

how are quadratic functions expressed?

Answer 17

y = ax² + bx + c

Question 18

what happens when “a” is positive?

Answer 18

the graph should have a smiley shaped curve

Question 19

what happens when “a” is negative?

Answer 19

the graph will have a frowny shaped curve

Question 20

what is the root?

Answer 20

where the graph cuts x axis and y is 0

Question 21

how do you find the root?

Answer 21

x = -b + or - the square root of b² - 4ac/ 2a

Question 22

whats the equation for revenue?

Answer 22

price x quantity sold

Question 23

what is the profit formula?

Answer 23

revenue - cost

Question 24

when do we find the breakeven point?

Answer 24

when profit = 0
when revenue = cost

Question 25

how to find breakeven on a graph

Answer 25

1 find the values of costs and revenue on each x point
2 plot both lines and breakeven is when the two lines meet

Question 26

what form is an exponential function in>

Answer 26

y = b^x

b is the base
x is the power/ exponent and it tells us how many times b multiplies by itself

eg y = 2^x and x = 3 then y = 2^3 and that equals 222 and that equals 8
can also be done for negative numbers

Question 27

in an exponential function what happens when y increases or decreases?

Answer 27

y increases by a constant factor with each unit of x
exponential function should grow faster

Question 28

what does a growth and decay function look like for exponential?

Answer 28

growth = curve up to right
decay = start from top left curve down

Question 29

example of exponential functions - finance (compound interest)

Answer 29

A = Pe^rt

A is the final amount
P is the principle amount
e is the base
r is the annual interest rate
t is time in years

Question 30

the sales of a product in year 0 are 200 units and are increasing by 10% each year what are the sales in year 1 and year n? if sales decrease by 15% what are the sales in year n?

Answer 30

y = b^x

year 0 = 200 units
10% increase = x 1.1
base is 1.1
yrn = 200 x 1.1^n

15% decrease = x 0.85
base is 0.85
yrn = 200 x 0.85^n

Question 31

what is a logarithmic formula?

Answer 31

reversing the process to find x so instead of 32 = 2^x we use the log in our calculator and log2,2^x as x is the base which will give you x by itself and log2,32 and that will give you the value of x

if you have a square the inverse is to square root

the inverse of e^x is log e,e^x or ln instead of log e on your calculator

Question 32

what is log (xy) equal to?

Answer 32

log(x) + log(y)

Question 33

what is log (x/y) equal to?

Answer 33

log(x) - log(y)

Question 34

what is the log (x^k) (log x to the power of k) equal to?

Answer 34

klog (x)

Question 35

what values will always give x as the answer?

Answer 35

10^log(x)
e^ln(x)
log(10^x)
ln(e^x)

Question 36

what is th opposite of log e/ ln?

Answer 36

e^x