Welcome ladies and gentlemen!
Today I want to explain to you something about zeros and intercepts for seven common types of functions. Those functions are the following: These 7 function types (you see the number of minnutes for each of the functions behind them, ie if you are interested in a particular function type , then you can switch to a certain point in time in the video. We start with linear functions. I show you an example. This linear function: The function term here is 8-4x. Where
are the the zeros and the intercept? You can calculate that. If you look at the equation, then the zero is the point at which the function becomes zero. For which “x” does the function become zero? To do this, we simply set the function to zero. y will become “0” if 8-4x=0. In order to calculate that, I change the equation to x. How is the interecept calculated? For this we insert x=0 into the original function. Of course, when we use zero, we yield 8 as the intercept. We mark these two points in our figure. Zero at 2 and intercept at 8. For linear functions, the calculations are still quite simple, but for quadratic functions, it will become a little bit harder So
I have a quadratic function as follows… This function is apparently shifted by 2 to the right and 1 down. The function term is therefore y=(x-2)^2-1 Until now, we do not know the zeros and the intercept, which is the intersection with the y-axis.
However, we can calculate that. If you want to calculate the zero, set the function=0 again. That is, one looks for the point where the function becomes zero. To calculate that, you need
the second binomial formula. In order to solve a quadratic equation we need the PQ formula. We’ll just insert p and q. The result is then 2 minus
Root 1, that is 1. These are the two zeros. How can you now calculate the intercept? You just have to insert x=0 in the
equation. Now we can see the zero and the intercept in the figure. These were the quadratic functions. Let’s switch to
cubic functions. This is an exemplary cubic function. Cubic functions have a
curved course. Here you can see this function. The equation of the function is x^3+2x^2-2x. One zero can be guessed, as it is at x=0. But of course we want to calculate
rather thang guess the zeros here. For this we now have to set the function equation to zero again. Here, the equation can be simplified by putting one x outside the brackets. So I’m excluding x from the same. Then you immediately see the first zero. We have already seen this zero in the graph. We need to calculate the other zeros
from the quadratic equation. This is done using the PQ formula. If you do that, you get another zero. It is at x=1. Remains to calculate the intercept. For this we insert x=0 into the starting equation. The result is y(0)=0. Hence, the intercept is at zero. We now mark the zeros and intercept again in the graph. These were cubic functions. Next is rational functions. These type of functions encompasses functions such as y=1/x. But I brought a funtion to you which is a little bit more complicated: It is shifted 2 units to the right and one unit down. The function term here is y=1/(x-2)-1. As you can see, this function also has a zero,
and also an intercept. Even if they are in separate sections. Of course you can also calculate the zero and the intercept. Again, we set the function equal to zero. (Calculation) (Calculation) The zero is thus at x=3. The function has – as I said – also an intercept. To calculate this, we insert x=0 into the output function. (Calculation) The intercept is at -1.5. In our graphic I mark the zero at 3 and
the intercept at -1.5. The next function type are root functions! These have a very special course. Look at this figure. The root function that you see here is obviously shifted. More precisely, this function is shifted two units to the left and one unit down. Consequently, the function term is (…) This function also has a zero and an intercept. Both are calculated now. As usual, the function is set to zero first. (Calculation) (Calculation) Then, the intercept is calculated. (Calculation) Both points can now be transferred to our graphic. I mark the zero at x=-1
and the intercept at y=0.41. This brings us to the next function type: The exponential functions and that’s where I show you
a very special function. The e-function. Of course, the procedure can also be adopted to
all other exponential functions because “e” is just a number. I show you an e-function here. This function has also been shifted. The equation of the function looks a bit
more complicated … The function is apparently shifted 3/2 to the left. Now we move to the calculation of the zeros and intercept. First, the function must be set to zero. (Calculation) To isolate the exponent there is a little trick which is to calculate the ln (logarithm naturalis) on both sides of the equation. (Calculation) (Calculation) (Calculation) The result is about -0.81. The intercept is
obtained by inserting zero into the output function. This leads to an intercept of
about 2.48. Let’s mark both in our graphic. Zero at -0.81 and intercept at 2.48. Let’s move on to our last function type: Logarithmic functions. This function type is also known as the
inverse function of the exponential function. I show you an example. This function obviously has a zero
and an intercept. The function equation is ln(2-x), that is
it’s a logarithmic function, which has been moved two units to the left. As usual, the zero is calculated first. (Calculation) To isolate the term in Ln, e ^() must be taken on both sides. (Calculation) (Calculation) (Calculation) Then the intercept is calculated. (Calculation) Ladies and gentlemen, thank you very much for your attention. Today, I showed you how to calcuate zeros of seven common function types
and how to determine intercepts. Of course, this is how it works also
for every other functions. I wish you much success.
Thank you very much!