Non-linear Functions

Question 1

What is the general equation for a quadratic function and what does each variable stand for?

Answer 1

y = ax^2 + bx + c

a= shape of parabolic curve
c= vertical position, also whether to not it has no roots or 2 roots (y-intercept) 
x= horizontal position= whether it shifts to the left or right

Question 2

How can you use the quadratic formula to work out how many roots there are?

Answer 2

If b^2-4ac >0: Two distinct roots

If b^2-4ac =0: One root

If b^2-4ac

Question 3

How could you rewrite the quadratic equation once you know the roots?

What does each variable represent?

Answer 3

y = a(x - α)(x – β) as it satisfies when y=0

Y-intercept: aαβ
Turning point: Half way between two roots: (α+ β)/2
Parabola is CONCAVE UP if a>0
Parabola is CONCAVE DOWN if a

Question 4

With the power function, y=x^n, what happens when n=even or n= old

Answer 4

Even values of n= concave UPWARDS, with centre on 0

Odd values of n= negative y for negative x, while positive y for positive x and CROSSES the origin

Question 5

What is the degree of a polynomial?

Answer 5

It is the largest exponent of X it constants e.g. x= linear, x^2= quadratic, x^3= cubic, x^4= quartic

Question 6

How do you work out how many turning points it has?

Answer 6

An nth degree polynomial can have up to n roots and n-1 turning points

Question 7

What does an asymptote represent?

Answer 7

When some functions approaches but never meets a particular value= Asymptote representing this value

The limit of a function is this value

Question 8

What is a logistic function?

Answer 8

Quadratic curves that initially rise, level out and then drop off

They pass through the origin

Useful to model population growth and carrying capacity

Question 9

What is the Gordon-Schaefer oder?

Answer 9

It is the yield-effort curve that was developed for fisheries management

Equation:
Y=aE(1-E/ k)
Where Y is yield and E is effort

Aim is to maximise the effort to yield relationship in relation to costs (Y/C)

C=mE

Question 10

What is the Michaelis- Menten function?

Answer 10

Predicts the relationship of enzyme-catalysed reactions is:
v= Vmax x s / (Km + s)

V= rate of reaction (μM s-1)

s= the concentration of the subject (μM)

Vmax= The maximum rate of reaction for a given quantity of enzyme. This is the horizontal asymptote

Km= Constant equal to the substrate concentration producing a rate of reaction half as fast as the maximum rate

Question 11

What does the Lineweaver-Burk transformation do?

Answer 11

Transforms the Michaelis-Menten relationship into a linear one

Equation: 1 / v = (Km / max x (1 / s) + 1 / max

c=1/Vmax
X-axis is crossed at -c/m= -1/Km