If you take a graph and transform it in various ways, here is what you get:
A reflection in the y-axis turns a graph of y=f(x) into the graph of y= f(-x).
A reflection in the x-axis turns a graph of y=f(x) into the graph of y= -f(x)
- note the difference -
A stretch of factor s in the y- direction turns the graph of y= f(x) into the graph of
y = s f(x).
A stretch of factor m in the x-direction turns the graph of y = f(x) into the graph of \[y = f(\frac{x}{m})\]
A translation by \[ \left( \frac{k}{l} \right) \] turns the graph of y = f(x) into the graph of y = f(x-k) + l
Tuesday 26 April 2011
Another Test
I'm trying to see if I can use LaTeX to write formulae onto the blog, but I've always been a bit befuddled by it. Instead I've installed a javascript thing from www.watchmath.com and that's supposed to recognise code and turn it into proper maths.
Well here goes.
Let's have a look at the function f(x) = 1/x. By rights in LaTeX that should be \[ \frac{1}{x} \]
Well here goes.
Let's have a look at the function f(x) = 1/x. By rights in LaTeX that should be \[ \frac{1}{x} \]
Test
\[ E(\mathbf{x}) = \sum_{i \in \mathcal{V}} \theta_i(x_i) + \sum_{ij \in \mathcal{E}} \theta_{ij}(x_i, x_j) \]
C3 Functions Round-Up
Reminder: this is really about AQA C3, as OCR and Edexcel are slightly different. They don't change the truths of mathematics though.
After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):
1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all.
2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).
3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x2 + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x2+4)-3.
4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.
After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):
1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all.
2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).
3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x2 + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x2+4)-3.
4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.
Wednesday 6 April 2011
Polynomials
These are a very common sight in almost any study of mathematics. It’s worth taking a moment to define them properly and think about how they can be used.
A polynomial is a mathematical expression involving a whole, positive power of x. The type of polynomial depends on the highest power of x in the expression. The different bits of the expression are known as terms, and the polynomial as a whole is the sum of the terms it contains.
Take a quadratic expression: 4x² + 2x - 2. It has three powers of x: 2, 1 and 0. These are whole numbers and positive (except 0). This is therefore a polynomial. The same applies to cubic expressions: x³ + 4x² + 5x + 2 is a polynomial too.
In fact, the same applies, whatever the power of x, as long as it is not a fractional or negative number. A polynomial does not need to have several terms, like most cubics and quadratics do. So, 1 is also a polynomial, being a term in x raised to the power of zero. The same applies to any constant (ie any number). 4x + 1 is a polynomial too, as x is x raised to the power of one.
Polynomials are written in descending powers of x, though they do not have to be.
The numbers in front of the x variables are called coefficients, and can take any value, including fractional or negative values, as can constants.
The highest power of x in a polynomial defines the degree of that polynomial. So 1 is a polynomial of degree 0, because the highest power of x here is 0. 4x + 1 is a polynomial of degree 1, and quadratics are polynomials of degree 2.
Polynomials can be created by expanding brackets. (x+2)² gives x² + 4x + 4 when expanded. In fact, both expressions here are polynomials. If the brackets were to give us an expression with a fractional power, then of course this would not be the case.
When you make a polynomial like a quadratic equal to 0, then you have a polynomial equation. In the case of a quadratic, such as 4x² + 2x – 2, the resulting equation is usually written as: 4x² + 2x – 2 = 0. As an equation, it is now a description of a parabolic curve which intersects the x-axis at (-1,0) and (½, 0).
Polynomials can be added, subtracted, multiplied and divided. The Factor and Remainder Theorems can be used to find out additional information about them. For example, if a number assigned to the variable x causes the polynomial to equal zero, that number is a factor of the polynomial (a root, a value of x where the curve of the equation crosses the x-axis). So we can demonstrate that -1 is a factor of 4x² + 2x – 2 by putting it in place of x. 4(-1)² + (-1x2) – 2 = 0. Therefore -1 is a factor of this polynomial. This is the essence of the Factor Theorem. The Remainder Theorem is a little more involved but states that if we divide the polynomial by x-a then the remainder is f(a) – in other words, the remainder is the sum of the polynomial when a is put in place of the variable x.
These are the absolute basics. Of course it does get a lot more complicated! But it is worth familiarising yourself with these principles before moving on.
A polynomial is a mathematical expression involving a whole, positive power of x. The type of polynomial depends on the highest power of x in the expression. The different bits of the expression are known as terms, and the polynomial as a whole is the sum of the terms it contains.
Take a quadratic expression: 4x² + 2x - 2. It has three powers of x: 2, 1 and 0. These are whole numbers and positive (except 0). This is therefore a polynomial. The same applies to cubic expressions: x³ + 4x² + 5x + 2 is a polynomial too.
In fact, the same applies, whatever the power of x, as long as it is not a fractional or negative number. A polynomial does not need to have several terms, like most cubics and quadratics do. So, 1 is also a polynomial, being a term in x raised to the power of zero. The same applies to any constant (ie any number). 4x + 1 is a polynomial too, as x is x raised to the power of one.
Polynomials are written in descending powers of x, though they do not have to be.
The numbers in front of the x variables are called coefficients, and can take any value, including fractional or negative values, as can constants.
The highest power of x in a polynomial defines the degree of that polynomial. So 1 is a polynomial of degree 0, because the highest power of x here is 0. 4x + 1 is a polynomial of degree 1, and quadratics are polynomials of degree 2.
Polynomials can be created by expanding brackets. (x+2)² gives x² + 4x + 4 when expanded. In fact, both expressions here are polynomials. If the brackets were to give us an expression with a fractional power, then of course this would not be the case.
When you make a polynomial like a quadratic equal to 0, then you have a polynomial equation. In the case of a quadratic, such as 4x² + 2x – 2, the resulting equation is usually written as: 4x² + 2x – 2 = 0. As an equation, it is now a description of a parabolic curve which intersects the x-axis at (-1,0) and (½, 0).
Polynomials can be added, subtracted, multiplied and divided. The Factor and Remainder Theorems can be used to find out additional information about them. For example, if a number assigned to the variable x causes the polynomial to equal zero, that number is a factor of the polynomial (a root, a value of x where the curve of the equation crosses the x-axis). So we can demonstrate that -1 is a factor of 4x² + 2x – 2 by putting it in place of x. 4(-1)² + (-1x2) – 2 = 0. Therefore -1 is a factor of this polynomial. This is the essence of the Factor Theorem. The Remainder Theorem is a little more involved but states that if we divide the polynomial by x-a then the remainder is f(a) – in other words, the remainder is the sum of the polynomial when a is put in place of the variable x.
These are the absolute basics. Of course it does get a lot more complicated! But it is worth familiarising yourself with these principles before moving on.
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