graphs and functions Flashcards
what is a function?
a mathematical process that uniquely links one or more variables to another variable eg advertising to sales
what does a function always have?
an input (domains)
a rule for processing the input(s)
an output
what are some real world functions?
business profit calculation: revenue- costs = profit
finance loan interest: principle amount we borrow x interest rate x time
what makes a function a function?
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
difference between x and y?
x is along the corridor y is up the stairs
how would you know if something was a function of something else?
if y = 0.4x + 50z then y is a function of x and z or in short y = f(x, z)
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
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
what is the linear function?
y = ax + b
gradient = a
intercept = b
in a linear function what happens when y increases or decreases?
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
how do you find the gradient?
(change in y) - (change in x)
what is the intercept?
point ar which function cuts the vertical axis
what is the gradient?
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
when drawing a function what must we consider?
put it into the function of y = ax = b to find y when x = certain points on your graph
how to find the equation of the line?
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 .
is a linear function for straight or curved lines?
straight
why cant we use a straight line to join the points on a curved graph?
because it would be inaccurate
how are quadratic functions expressed?
y = ax² + bx + c
what happens when “a” is positive?
the graph should have a smiley shaped curve
what happens when “a” is negative?
the graph will have a frowny shaped curve
what is the root?
where the graph cuts x axis and y is 0
how do you find the root?
x = -b + or - the square root of b² - 4ac/ 2a
whats the equation for revenue?
price x quantity sold
what is the profit formula?
revenue - cost
when do we find the breakeven point?
when profit = 0
when revenue = cost
how to find breakeven on a graph
1 find the values of costs and revenue on each x point
2 plot both lines and breakeven is when the two lines meet
what form is an exponential function in>
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
in an exponential function what happens when y increases or decreases?
y increases by a constant factor with each unit of x
exponential function should grow faster
what does a growth and decay function look like for exponential?
growth = curve up to right
decay = start from top left curve down
example of exponential functions - finance (compound interest)
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
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?
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
what is a logarithmic formula?
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
what is log (xy) equal to?
log(x) + log(y)
what is log (x/y) equal to?
log(x) - log(y)
what is the log (x^k) (log x to the power of k) equal to?
klog (x)
what values will always give x as the answer?
10^log(x)
e^ln(x)
log(10^x)
ln(e^x)
what is th opposite of log e/ ln?
e^x
