• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Math assignment - Families of Functions.

Extracts from this document...

Introduction

Erik Skyllberg                                                NEG

2003-04-03

Math assignment

Families of Functions

Linear Function:

image00.png

A linear function is formed by the slope-intercept form y = mx + b, where m is the gradient/slope, and b is the y-intercept. The greater gradient, the steeper the line will be.

The gradient can be calculated through taking ∆y/∆x. Then you will have to use two points on each line.

The y-intercept of my first linear function graph, the red line, is 2, due to that b = 2. The gradient is + 3, which also can be written 3/1. To find out that, it is possible to use the rise/run method. That means that you already know one point on the graph, the y-intercept + 2. From that point you can go upwards (rise) three units, and one unit to the right (run). You could also use the ∆y/∆x method to calculate the gradient. Then you take two co-ordinates on the graph. In my example I have chosen to take (-1, -1) and (0,2). I then take (2-(-1)) / (0-(-1)) = 3/1 which gives me the same answer (y2-y1 / x2-x1).

In my second linear function graph, the gradient is negative, which means that my line will be moving downwards from left to right.

...read more.

Middle

image02.png

Quadratic functions are written in the standard form y = ax2 + bx + c. This type of graph is like a parabola that opens upward, or downward. If a < 0, the parabola opens downward, and if a > 0, the parabola opens upward. The graph will be more narrow the greater the value of a is. So, as you can see in my graph, the value of a is greater in my first graph, than in my second graph, and the parabola is therefore more narrow.

The axis of symmetry can be calculated through taking x = (-b) / (2a), so in my case for the first graph, it would be x = (-2) / (2*3) which equals –⅓ which you can see is correct. The axis of symmetry is used to see that the graph is correctly drawn. You can fold the paper there, and the graph should look exactly the same on both sides, if you have done it right.

The vertex, which has the maximum value of the graph if it opens downward, and minimium value of the graph if it opens upward, ( -b / (2a), y). In the first of my graphs, it would be

( (-2) / (2*3), -(2 ⅓) ).

The y-intercept in a quadratic function is always the value of c.

...read more.

Conclusion

x + 3 ≥ 0, which gives us x ≥ -3.The domain is now set to be all numbers that are grater than or equal to –3. The starting point of the graph is though the point of ( b , c), and so if there is no c value, the y value will be 0. It shifts horizontally with the b units, and vertically with the c units.

Rational Functions:

image05.png

A rational function can be written in the form y = polynomial / polynomial. A polynomial is one term or the sum or difference of two or more terms. The value of the variable cannot make the denominator equal to 0, so if I would have used y = x / (x – 4), x ≠ 4.

My first function is an example of a discontinuous graph is, since the variable is in the denominator, y is not defined for x = 0. When you look at the graph for this function,

you will see an asymptote at x = 0, which is the y-axis. When you try to sketch this graph, you may plot a few points and then attempt to connect them into one curve. However, as you approach zero from the negative direction, you will have to stop at x = 0, lift your pencil

and start again at the lowest positive number.

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    this proves that proving it binomially is equal to the increment method. When next using other values of n, I shall present a table of results when next picking two points, unlike when previously illustrated in the increment method for y =x�.

  2. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    Thus, assume that consumption towards their consumption to their long-term goal. Then, consumption function in this case is: ?ct=c0+c1?yt+c2st-1+c3?t Use the same ways to gain the test result to estimate the consumption function. The following consumption function can be generate from the data is: DLC=+0.01119+0.6508*DLY+0.1092*LS_1-0.001078*p In Table 6, the result

  1. Investigate the number of winning lines in the game of connect 4.

    27=a5+b (2) 9=a (2)-(1) Substitute 'a' back into (1) 18=9x4+b 18=36+b b= -18 Substitute 'a' and 'b' back into second number equation S=9w-18 So we have the equations for the first and second numbers of the table. So now we put them together. The first number equation is solved by 4W-9 and then to that you had to times

  2. Exponential decay:The objective of this experiment is to get the trend line of heart ...

    Because some people ran fast, some people ran slowly. 3.0 Analysis 3.1 Experimental value From the graph, we can get the equation, 155.89 can be rounded to 156 which is the original number of heartbeat after excising. -0.0934 is the decay constant which represent . 3.2 Data Analysis This is the heart rate that our body will decrease to after an exercise session.

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    First of all, the graph is stretched vertically by a factor of 1.627. Afterwards, the graph is stretched horizontally by a factor of 0.016. After that, the graph is translated 111.744 units left. Lastly, the graph is translated 6.248 units up.

  2. The Gradient Fraction

    I begin to identify a pattern; the values on the 'y' and the 'x' scales are the same, so the gradient of the straight line with the same scale values will equal to 1. This is because when you divide the value on the 'y' axis by the value on

  1. Investigating the Quadratic Function

    eg: y = (x + b)� This is a perfect square where if the square root of 'x + b' was taken then the answer to 'y' would be established. Here are some graphed examples of perfect squares: D E F y = x� y = (x - 2)� y

  2. Fractals. In order to create a fractal, you will need to be acquainted ...

    With Mandelbrot?s set, we can determine the fate of the orbits. For example, when we substitute the constant, z0, as 1, and keep the seed of the iteration as 0: z1 = 02 + 1 = 1 z2 = 12 + 1 = 2 z3 = 22 + 1 =

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work