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Math assignment - Families of Functions.
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Introduction
Erik Skyllberg NEG
2003-04-03
Math assignment
Families of Functions
Linear Function:
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.
Middle
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.
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:
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.
This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.
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