How to find graph of a function ?
The task of building a graph of the students faced in the beginning of the study of algebra and continue to build them from year to year.Since the graph of a linear function, for the construction of a need to know only two points to the parabola, for which you need to have 6 points, hyperbole and a sinusoid.Every year, functions become more and build their schedules no longer possible to run the pattern, it is necessary to carry out more sophisticated studies using derivatives and limits.
Let's see how to find the graph of a function?To do this, start with the most basic functions of graphics which are based on the points, and then consider the plan to build more complex functions.
Graphing linear function
To construct a simple graph using a table of function values.The graph of a linear function is a straight line.Let's try to find the point of the graph of y = 4x + 5.
- For this we take two arbitrary values of variable x, is alternately substitute them into the function to find the value of y and is listed in the table all.
- take the value x = 0 and substitute the function instead of x - 0. We obtain: y = 4 * 0 + 5, that is, y = 5, we write this value in the table below 0. Similarly, we take x = 0, we obtain y = 4 * 1+5, y = 9.
- now to build a graph of the function must be applied on the coordinate plane, these points.Then you need to draw a straight line.
Graphing quadratic functions
quadratic function - a function of the form y = ax2 + bx + c, where x-variable, a, b, c - the number of (a is not 0).For example: y = x2, y = x2 + 5, y = (x-3) 2, y = 2x2 + 3x + 5.
To construct the simplest quadratic function y = x2 usually take 5-7 points.Take the value for the variable x: -2, -1, 0, 1, 2 and find the value of y as well as the construction of the first schedule.
graph of a quadratic function is called a parabola.After charting function pupils there are new challenges associated with the schedule.
Example 1: Find the abscissa of the point of the graph of y = x2, if the ordinate is equal to 9. To solve the problem must be in the function instead of y we substitute its value obtain 9. 9 = x2 and solve this equation.x = 3 and x = -3.This can be seen on the graph of the function.
study function and construction timetable
charting more complex functions requires a few steps to her research.To do this:
- Find the domain of the function.Area definition - all values that can take the variable x.From the domain of definition should exclude those points where the denominator is equal to 0 or a radical expression becomes negative.Set
- even or odd function.Recall that even that is a function that satisfies the condition f (-x) = f (x).Its graph is symmetric about Oy.The function is odd if it meets the condition f (-x) = - f (x).In this case, the graph is symmetrical about the origin.
- Find the point of intersection with the coordinate axes.To find a point of intersection with the abscissa of the x-axis, it is necessary to solve the equation f (x) = 0 (when this ordinate is 0).To find the y-axis ordinate of the intersection point, instead of the necessary function of the variable x to substitute 0 (abscissa axis is 0).
- Find asymptote function.Asiptota - straight, to which the graph infinitely close, but it never crosses.Let's see how to find the asymptote of the graph of.
- vertical asymptote line of the form x = a
- Horizontal asymptote - a direct form y = a
- Inclined asymptote - a direct type y = kx + b
- Find point of extremum, increasing intervals and descending function.Let us point extremum.To do this, find the first derivative and equate it to 0. It is at these points, the function may change from ascending to descending.Define the sign of the derivative at each interval.If the derivative is positive, the graph of the function increases, if negative - decreases.
- Find the schedule of inflection points of a function, convexity intervals up and down.
Find the inflection point is now easier than ever.We just need to find the second derivative, and then equate it to zero.Following find a sign of the second derivative at each interval.If positive, the graph of the function is convex downward, if negative - up.