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At turning points, the gradient is 0. Differentiating an equation gives the gradient at a certain point with a given value of x. To find turning points, find values of x where the derivative is 0.
Thus, there is on turning point when x=5/2. To find y, substitute the x value into the original formula.
Thus, turning point at (5/2,99/4).
Once turning point is identified, you can work out if it is a maximum or minimum by finding d2y/dx2.
d2y/dx2<0 - maximum
d2y/dx2.>0 - minimum
Thus for our example above
d2y/dx2=2 - minimumsee more
A quadratic equation is one that includes x2 as the highest power of x. Factorising is achieved in 3 steps. Let’s consider the example x2-3x-3=1
1) Put the equation into the form ax2+bx+c=0
We need two numbers that
- add together to get -3
- Multiply together to get -4
-4x1=-4 and -4+1=-3
Thus, factorising gives (x-4)(x+1)=0
3) Solve the equation!
If two numbers are multiplied together to give 0, one of them must be 0. Thus:
x-4=0 and x=4
x+1=0 and x=-1
The equation has been solved
- This technique can be applied to finding the points of intersection on the x axis for a quadratic graph. For example, y=x2-3x-4. At the x axis, y=0 so you can work out x as above.
- Harder quadratic equations can also be solved by factorising. For example when a isn't 1.
2x2 + 7x + 3=0
Find two numbers that multiply to give 2x3 (6) and add to give 7. In this case, 6 and 1.
Split 7x into 6x +x
2x2 + 6x+x + 3=0
Factorise each part by taking out a common factor.
2x(x+3)+1(x + 3)=0
The sames as
thus x = -1/2 or x=-3
1. Solve by factorising
x2 + 6x + 8=0
x2 – 8x + 16 = 0
2. Find the points of intersection with the x axis for
y=x2 – 14x + 48
and sketch this functionsee more
Meiosis produces 4 haploid daughter cells and is essential for forming gametes that contain one copy of each chromosome. Meiosis allows genetic diversity which gives rise to new combinations of genetic alleles so that offspring have varied phenotypes and natural selection occurs.
Genetic diversity is achieved by 3 main mechanisms:
1) Random chromatid assortment
When paired homologous chromosomes line up on the equator in metaphase 1, paternal and maternal chromatids are assigned to either cell randomly. Thus, each cell receives a random combination of maternal or paternal chromosomes.
2) Crossing over
In order to keep homologous chromosomes paired, chiasmata form in which complementary sections of homologous chromosome cross over thus exchanging genetic material. This creates new allele combinations on the same chromosome. The further the genes are from the centre of the chromosome, the easier crossing over is and the more frequent recombination.
3) Random fertilisation
Any sperm can fertilise any ovum and thus again, a random combination of homologous chromosomes is produced in the zygote adding to genetic diversity.see more