- Edexcel - C1, Algebra and Functions
- AQA - C1, Algebra
- OCR - C1,Coordinate Geometry and Graphs
Almost finished with Algebra and Functions, We now need to know how to sketch certain functions, aswell as know how to interpret graphs to solve equations, and know how to apply different transformations (this will be covered in the next post).
In the specification, it says sketching curves defined by simple equations. I shall teach you how to draw linear, quadratic and cubic functions. Though the first two you should have known from gcse. Aswell as teaching you k/x functions where k is a integer (this is a reciprocal function).
Linear Functions
So if you see a function like y=3x + 2, or y= x or y = -4x.... these are known as linear functions, simply because they are straight lines.. and are of the power 1 (exponent). We usually see them in the form y=mx + c, but we will see in the next section Coordinate Geometry, they can be rearranged in the form ax + by + c aswell. Let's focus in the form y= mx + c.
m = gradient of the function (slope)
c = y intercept
If you take y to be 0 : you will get 0 = mx + c, rearrange for x this should be the point on the function when y=0, or where it crosses the x-axis. Likewise if you make x=0, you will get y=c, this is the point on function when x =0, or where it crosses the y-axis.
To plot a function, if you take different values of x and y, you could plot it up... though we are meant to sketch functions in a general way. So Linear Functions are straight lines .. so let's take 3 examples.
a) y= 2x - 5
Firstly we know m = 2 (gradient), and c = -5 (y intercept), this is easy to see in this form, but not all linaer functions are in this form REMEMBER, y= 4x , is still a linear function even though c = 0.
y intecerpt = take x =0, so y= 2(0) - 5
y= -5
We've got a coordinate on the function which is (0, -5)
x intercept = take y =0, so 0 = 2x - 5 so 5=2x so x = 5/2
Another coordinate is (5/2 , 0)
As we know it is a straight line just draw a corresponding straight line using the two points :
Graph of y=2x-5, notice where it croses the axis, it's just a straight line through the points |
b) Let's take y = -6x
Here m = -6 (negative gradient ), and c = 0
y - intercept, y = -6(0), y = 0, so when x =0, y= 0.
First coordinate (0,0)
x- intercept, y =0 so 0 = -6x so 0/-6 = x, so x =0
Second Coordinate (0,0)
As we already knew (0,0) from the first coordinate, the 2nd step was futile.. so we
simply substitute other values to get a general shape of the graph.
when x = 1, y= -6
when x = -5, y = 30
Graph of y = -6x, notice the gradient (m) is negative, so it is downward sloping, and it goes through the origin (0,0). |
Quadratic Graphs
As we know from the previous section, that quadratic functions are in the form ax^2 + bx + c = 0. If we have two solutions to the equation (two roots), these are the corresponding x - values where y =0, or the two x intercepts.
We know that a quadratic is a parabola shape, and the coefficient of x squared, determines whether it is a U or a upside down U. If the coefficient is negative, it is an upside down U, if positive it is a U shape. Again we know c is the y intercept ( or where x =0). There will also be a minimum or maximum point on the graph, (depending the parabolic nature), this can be found through differentiation (later on...).
e.g. 1 ) y = 2x^2 - 8x + 6
Firstly a =2, so it will be a U shaped parabola shape.
substitute x =0, so y= 6, the y intercept is 6. One coordinate is (0,6)
Now find the roots of the quadratic, (through one of the three methods).
Let's do it by factorisation, divide everything by 2.
x^2 - 4x + 3 = 0
(x-3)(x-1) = 0
so x=1,3 (these are the points which the function cross the x-axis.
(1,0) and (3,0)
Graph of example above. a) positive a, so U shape parabola, and two roots. |
2nd example :
y = -x^{2} - 4x - 3
a is -1, so it is an upside down U shape.
y intercept is -3 , (0,-3)
now solve for x :
-x^{2} - 4x -3 = 0
-(x -1) (x-3) =0
so x =-1 and x = -3
so (-1,0) and (-3,0)
Graph of -x^{2} - 4x -3, notice where it crosses the axis and the shape. |
Cubic Functions
A cubic is in the form ax^3 + bx^2 + cx + d = 0, here again the a decides the shape... we know the cubic shape is a kind of S shape (looking at it with your head tilted). if a is negative then the graph will go from right to left, if positive, then left to right. Here's what i mean :
When a is positive |
When a is negative |
Again we have to find the x intercepts, by setting the equation equal to 0. And solving, this can be done sometimes if you know one factor, leaving you with a quadratic, which would give you the other two. The y-intercept is found by making x=0.
A cubic will have either 3 roots, 2 roots or 1 root. (it crosses the x-axis 3 times, 2 times or once.)
Graph of x^3, notice here there is one root, at the origin (0,0). |
Example 1
y = x(x-2)(x+3)
This is a cubic, but factorised, it we were to expand it, it would give us in the form of ax^3 + bx^2 + cx + d, but when factorised it is easy to draw a cubic. We are only expected to sketch easy cubics at this stage, as solving it, is in C2 (using factor theorem, and algebraic division). If it is already factorised, it is easy to draw... for this example. We have three roots, x=0 , x =2 and x =-3, these are the points where the function cross the x-axis.Remember, so cubics can have two roots, so one may be a repeated root.
Graph of the cubic (x^3 + x^2 - 6x), when expanded, notice the roots at -3,0 and 2. It has two turning points. |
x^3 -9x.
Let's try factorising this, if we take x out we get x(x^2 -9). So one root is 0, now we have x^-9. Remember a difference of two square where you have something in the form of x^2 - a. Where a is a square number, if this is the case, the roots of the equation are +root of a and -root of a. In this case a = 9, so the roots are 3 and -3.
All the roots of this cubic are -3, 0 , 3. So we have the coordinates (-3,0) (0,0) and (3,0). The y intercept is 0.
Three roots at 0, -3 and 3. |
Reciprocal Graphs
The reciprocal function is y = 1/x. The curve of this function looks like a hyperbola, this is the graph of 1/x :
Graph of 1/x |
Assymptote - A line where a curve gets closer and closer to but does not meet.
By substituting 0 as x or y, we can find the intercepts of the axes.
If two graphs meet, this algberically, means setting two of the equations equal to each other, the coordinates of intersection, can be found by solving.
A cubic can have one, two or three roots.
Thanks :) This is very helpful!
ReplyDeleteWas wondering if you could do a post on sequences and series?
Thanks again for taking your time out on Christmas for this :D
Glad to know that you found the post helpful, I shall finish Algebra and Functions tomorrow. Coordinate Geometry (the next section) will be done by Wednesday. I shall start Sequences and Series on Thursday :)
ReplyDeleteI just like the helpful information you provide on your articles.
ReplyDeleteI will bookmark your weblog and take a look at again
right here regularly. I'm fairly sure I will be informed lots of new stuff proper right here! Best of luck for the next!
Review my weblog - Wholesale jerseys Cheap
Informative article, just what I was looking for.
ReplyDeleteAlso visit my web page; Louis Vuitton Bags
I know this if off topic but I'm looking into starting my own weblog and was curious what all is required to get set up? I'm assuming having a blog like yours would cost a pretty penny?
ReplyDeleteI'm not very internet savvy so I'm not 100% sure. Any tips or advice would be greatly appreciated.
Kudos
Feel free to surf to my webpage :: Louis Vuitton Outlet
Howdy! This post could not be written any better!
ReplyDeleteReading through this post reminds me of my good old room mate!
He always kept talking about this. I will forward this article to him.
Fairly certain he will have a good read. Thanks for sharing!
my page check this out
Thanks for finally writing about > "Graphs of Functions and Interpreting Graphs" < Loved it!
ReplyDeleteMy weblog ... Clicking Here
I loved as much as you will receive carried out right here.
ReplyDeleteThe sketch is tasteful, your authored subject matter stylish.
nonetheless, you command get got an nervousness over that you wish be delivering the following.
unwell unquestionably come further formerly again since exactly the same nearly very often inside case you shield this increase.
My website ... Michael Kors Canada - -
It's really a cool and useful piece of information. I'm glad that you shared this useful info with us.
ReplyDeletePlease keep us up to date like this. Thank you for sharing.
my web-site; Mulberry Bags
It's appropriate time to make a few plans for the longer term and it is time to be happy. I have learn this publish and if I could I desire to suggest you few interesting issues or tips. Maybe you can write next articles relating to this article. I desire to read even more issues approximately it!
ReplyDeleteLook at my page Christian Louboutin Outlet - -
Hi there! I just wanted to ask if you ever have any trouble with hackers?
ReplyDeleteMy last blog (wordpress) was hacked and I ended up losing
months of hard work due to no backup. Do you have any solutions to protect against hackers?
Feel free to visit my blog :: NFL Jerseys Cheap
great article, I was very impressed about it, wish you would have stayed next share
ReplyDeletekids games , friv2 , jogos de friv , juegos.com , juegos de matar zombbies
, juegos de un show mas
This article is really fantastic and thanks for sharing the valuable post.
ReplyDeleteSignature:
i like play games happy wheels online and play happy wheels 2 games and Download retrica online includes more than eighty different filters with many different styles and include retrica indir , and zombie tsunami is the ideal game for anyone who loves the running game genre
ugg boots, longchamp outlet, louis vuitton, christian louboutin, louis vuitton outlet, ray ban sunglasses, louis vuitton outlet, uggs on sale, gucci handbags, christian louboutin uk, tiffany jewelry, chanel handbags, louboutin pas cher, air max, nike free run, prada handbags, louis vuitton outlet, jordan pas cher, louis vuitton, longchamp pas cher, oakley sunglasses, polo outlet, replica watches, nike free, nike roshe, kate spade outlet, nike outlet, tiffany and co, polo ralph lauren outlet online, ray ban sunglasses, nike air max, oakley sunglasses, jordan shoes, ugg boots, oakley sunglasses wholesale, prada outlet, nike air max, longchamp outlet, longchamp outlet, tory burch outlet, michael kors pas cher, sac longchamp pas cher, oakley sunglasses, ray ban sunglasses, replica watches, burberry pas cher, christian louboutin outlet, cheap oakley sunglasses, polo ralph lauren
ReplyDeletetimberland pas cher, michael kors outlet, nike free uk, guess pas cher, michael kors, michael kors outlet, hollister uk, ralph lauren uk, abercrombie and fitch uk, north face uk, coach purses, burberry outlet, converse pas cher, coach outlet, polo lacoste, nike blazer pas cher, uggs outlet, michael kors outlet online, michael kors outlet online, hollister pas cher, lululemon canada, michael kors, sac hermes, true religion outlet, uggs outlet, nike air force, michael kors outlet online, burberry handbags, sac vanessa bruno, michael kors outlet, nike air max uk, new balance, true religion outlet, mulberry uk, vans pas cher, ray ban uk, ray ban pas cher, nike tn, true religion outlet, kate spade, north face, nike air max, true religion jeans, coach outlet store online, michael kors outlet online, hogan outlet, oakley pas cher, nike air max uk, nike roshe run uk
ReplyDeleteinsanity workout, hollister clothing, jimmy choo outlet, longchamp uk, soccer shoes, iphone 6s cases, ralph lauren, hermes belt, iphone 6 cases, abercrombie and fitch, mont blanc pens, timberland boots, s6 case, mcm handbags, chi flat iron, lululemon, mac cosmetics, giuseppe zanotti outlet, iphone 6s plus cases, ferragamo shoes, herve leger, iphone cases, celine handbags, north face outlet, nfl jerseys, valentino shoes, soccer jerseys, iphone 5s cases, wedding dresses, bottega veneta, nike huaraches, ghd hair, baseball bats, iphone 6 plus cases, north face outlet, p90x workout, instyler, vans outlet, new balance shoes, oakley, ipad cases, nike trainers uk, babyliss, louboutin, beats by dre, reebok outlet, nike roshe run, nike air max, hollister, asics running shoes
ReplyDeletelancel, swarovski, hollister, swarovski crystal, moncler uk, juicy couture outlet, montre pas cher, doudoune moncler, links of london, ray ban, ugg uk, moncler, louis vuitton, moncler, hollister, moncler outlet, moncler, barbour uk, louis vuitton, replica watches, nike air max, louis vuitton, toms shoes, barbour, canada goose outlet, converse outlet, canada goose outlet, marc jacobs, louis vuitton, ugg, juicy couture outlet, pandora jewelry, wedding dresses, canada goose, louis vuitton, ugg,uggs,uggs canada, pandora charms, canada goose uk, pandora uk, pandora jewelry, moncler, canada goose outlet, gucci, moncler outlet, ugg pas cher, converse, thomas sabo, karen millen uk, ugg,ugg australia,ugg italia, canada goose, supra shoes, canada goose jackets, vans
ReplyDelete