Absolute Value Function:
In an absolute value function, the graph is determined from the sign that is before the absolute value signs, since what is inside always will be positive after removing the absolute value signs. The characteristics of an absolute value graph are that it is a V-shape, either positive that opens upwards, or negative that opens downwards, as shown in my examples. The function is often written Y = | x - b|. The vertex is (b, 0). Since the b is inside the absolute value symbols, it is always positive, but there can be a negative sign before those absolute symbols, that change it, as in my second function.
The y-intercept for an absolute value function is b, and it is not to be forgotten that b is inside the absolute value symbols. So, you calculate the graph through using the three points that you have got, the y-intercept, the vertex, and the point, symmetric to the y-intercept.
Quadratic Functions:
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. The easiest way to draw a quadratic function is through using a Graphing Calculater, or a table of values where you enter the domain and range.
Quadratic functions can, but do not have to have a solution. The solutions are the x-intercepts, and can easily be calculated with the formula x = (–b ± √ (b2 –4ac)) / 2a, and that will give us the solutions. You have to use this formula twice, once with the positive sign before the square root of, and once with the negative sign.
You can also use the discriminant to tell weather there are one, two or none solutions, as shown in the table:
Exponential function:
For the exponential function the form y = abx is used, where b shows us a growth if it > 1, and a decay if 0 < b < 1 and b ≠ 1. This kind of function is often used for growth or decay during days, months, years etc. The y-intercept is always a, and is the original amount that you have, for example if we would say that my first graph would be my money on a bank account in thousands. I have 3000 kr from the beginning, and it is a very good account, so I will get 20% interest every year, but the time might vary, thus in other case it might be every month or minute. The value of b is 1.2, since that is the percentage increase in decimal form. The x-scale in this case represents the years, so from looking at my graph I can easily see that after 5 years I will have 7500 kr on my account. You need to do a table of values to draw an exponential function (or use a GCD). The points have to be connected with a smooth curve, since it is not a straight line.
In my second graph, the y-intercept is 8. This shows us decay, since the curve is negative. My initial amount was 8, and it decreased with 50% each period of time, so after just one year, the amount left was only 4, which you can see on the gray spot on the graph.
Radical Functions:
Radical functions are written in the form y = √(x - b) + c. C decides how many steps up or down on the y-scale the line will start.
To solve this type of equation, a table of value is needed. From that one you can see where the points are to be in the graph.
The domain for a radical function is very limited, because the expression under the radical cannot be negative. To find the domain for the second of my functions, you should solve
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.