if you never remember what was said in math class what does this video do because I'm going to tell you everything about the quadratic function and how to graph it in super detail so you understand it perfectly subscribe to the channel and we start we already saw that a function is the relationship that links the elements of a set with those of another set in mathematics we call them domain and image independent variable and dependent variable x and also x and fx each value of x is assigned a single value of fx this is a function I give a value and another is returned to me with this clear now we can go fully into the famous quadratic function we can start talking about the characteristics that we know if we look at its algebraic formula which is f x equaled by x squared plus b times x plus c this is made up of three terms and sometimes they can be any real number either a positive or negative any fractions a is the coefficient that multiplies x squared b the coefficient of x and a we call them independent term it does not matter if we find them in another order because the most important thing for a function to be is that the greatest number the one that the x is raised to the 2 and not the 3 to the 4 to the 5 so this is why it is said that it is a function of degree 2 or second degree then a can never be equal to 0 because if it were so we would not have the x raised to the square and it would be a function of first degree or a linear function on the other hand when we see the graph of a quadratic in the Cartesian plane we can find new characteristics when graphing it we will always have a parabola that will change shape according to the values of the function if it is positive the parabola will have the shape of smiles but if it is negative it will have the shape of a sad mouth the greater the absolute value of a the more closed the parabola will be and the lower it is the more its branches will open parabolas have the characteristic of being symmetrical and the vertex is their most characteristic point because it divides them exactly in half it can be their highest or lowest point and in the case of this function the vertex is one minus one and it is its lowest point this is the axis of symmetry which is the line that lived graphically also exactly by half in this function the axis of symmetry is x1 important points of the function are those that cross the coordinate axes a parabola will always cross the x axis in at most two points sometimes it may happen that it does not cut it or it cuts it but only once instead the y axis will always cut it yes or yes in a single point from the vertex we can say where the parabola grows and decreases if we see that as the values in x increase they also increase in y then the graph goes up or grows but if the values in x increase and those of iu decrease or the graph goes down then the curve is decreasing this function that we see grows from the interval minus infinity 1 and decreases in one plus infinity with the vertex is a point that does not grow or grow we do not include it in the intervals and we express them as open when looking at a function on the plane we can also talk about its domain the values it takes in x and its image the values that they take in the domain of a quadratic will always be the entire set of real numbers because although we have only a part of the function the word always opens as we move in x to both sides but to know the image of the function we must know its vertex because this determines the highest or lowest value it will take in and if the parabola is positive this will be its lowest point and that is why its image starts at the vertex guide to infinity but if it is negative the vertex is a higher point and then its image starts at minus infinity to the iv of the vertex the vertex if it belongs to this interval then we express it as a closed end when we have infinity remember that it is always expressed as an open end the domains of these two functions are all real numbers but the image of this function becoming one minus one is minus one infinity and the image of this other function with vertex at 12 is minus infinity 2 but now how do we graph the plane functions what we have to do is gather good information, that is, representative coordinates of our parabola to graph both where it goes up and where it goes down the first thing we have to do is be very clear about the values of a, b, and c in this function a is equal to two times equal to 4 and c is equal to 0 because it does not appear now we can find out its vertex for is the x of the vertex we obtain the following formula less than over 2 x a we replace the values we calculate and we have the x of l'or says is minus 1 to obtain it and of the vertex we only have to replace this x in the function we calculate and we obtain that it is minus 2 then our vertex is minus one minus two as the is positive the function of in a smile shape then this will be its lowest point the second step to graph is to look for the intersection points on the Cartesian axes at the intersection point on the x axis it will always be zero then if in our function we replace x by zero these two terms will disappear and we are left with that y is equal to zero and in the case of our function it is the zero point zero on the other hand the intersection points with the x axis which is also called sets of zeros will be the coordinates where y is equal to zero if we replace we have that zero is equal to 2 times x squared plus 4x this is an equation quadratic and we'll find its x by using the quadratic solvent network which is x is equal to minus plus minus the square root of minus 4 x to porsche over two times a thanks to this + - is that we're going to get two values for x we're going to replace all the values of our function and we calculate x is equal to minus 4 plus minus the square root of 4 squared minus 4 times 2 times 0 over 2 x 2 minus 4 times 2 times 0 is 0 and 2 times 24 16 minus 0 is 16 and its square root is 4 minus four plus negative 4 over 4 here we'll divide the account for each x minus 4 4 over 4 gives us 0 over 4 which is the same as 0 times the other minus four minus four over four is negative eight over four so the result is minus 2 x 1 is equal to 0 and x2 is equal to negative 2 now yes we can say that our quadratic cuts the x-axis by the points 0 0 and negative 20 finally we're going to make a table of values to add some extra coordinates that gives us a more precise graph the coordinates that we choose to find out do not have to make many just the necessary and the most coherent to complete our graph in this case I am going to look for them and for x equals minus 3 and x equals 1 because I am seeing that I am missing information to be able to extend the branches of my parabola we replace the x we calculate and now I know that others of its coordinates are minus 36 and 16 we can now join all the points and ball we have to graph our quadratic we can say that its axis of symmetry is x equals minus 1 its domain is all the real numbers its image is minus 2 infinity grows from minus 1 to infinity and decreases from minus infinity to minus 1 this was all you needed to know about the quadratic function if this video is useful to you do not forget to share it subscribe to the channel and ring the bell so you don't miss the next ones [Music]