LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

GRAPHS OF REAL FUNCTIONS

__GRAPHS OF REAL FUNCTIONS -__

**real functions can take many forms and shapes, depending on their equations** **and parameters. Here are a few examples of graphs of real functions:**

__Linear Function:-__The graph of a linear function is a straight line. It has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.__Example:-__f(x) = 2x + 3__Quadratic Function:-__The graph of a quadratic function is a parabola. It has the form f(x) = ax²+ bx + c, where a, b, and c are constants.__Example:-__f(x) = x²− 4x + 4.__Exponential Function:-__The graph of an exponential function is a curve that increases or decreases exponentially. It has the form f(x) = a⋅b^x, where a and b are constants.__Example:-__f(x) = 2x__T____rigonometric Function:-__Trigonometric functions like sine, cosine, and tangent produce periodic graphs.__Example:-__f(x) = sin(x).__Logarithmic Function:-__The graph of a logarithmic function is a curve that increases or decreases logarithmically. It has the form f(x) = logₑ(x), where e is the base of the logarithm.__Example:-__f(x) = log(x).__Sine Function:-__f(x) = sin(x),__Cosine Function:-__f(x) = cos(x),__Tangent Function:-__f(x) = tan(x).__Piecewise Functions:-__These functions are defined by different expressions over different intervals.

__The graphs of a function f consists of all points (x, y) such that y = f(x). To graph a function we carry out three steps:-__

__Step.1)__ Make a table of pairs from the function.

__Step.2)__ Plot enough of the corresponding points to learn the shape of the graph. Add more pairs to the table if necessary.

__Step.3)__ Complete the sketch by joining the points.

**Illustration. Suppose we have to draw the graph of function y = x² over the interval - 2 < x < 2.**

**We have graphed the function y = x² over the interval - 2 ≤ x ≤ 2.**

**The domain and range of y = x² are both infinite, so we can not hope to draw the entire graph. But we can imagine what the graphs looks like by examining the formula y = x² and by looking at the picture we already have. As x moves away from the interval -2 ≤ x ≤ 2 in either direction y = x² increase rapidly. When x is 6, y is 36. when x = 12, y is 144. Continuing the pattern we see that the graph goes up as shown in above figure.**

**The basic idea for graphing curves that are not straight lines is to plot points until we see the curve's shape, and then to use the formulae for y to find out how y changes as x moves between or away from the plotted values. But what points should we plot? Here are some guidelines about choosing good points to plot.**

**Guidelines for choosing points for graphing y = f(x):-**

**1) Plot any points where the graph crosses or touches the axes. These points are often easy to find by setting y = 0 and x = 0 in the equation y = f(x).**

**2) Plot a few points near the origin. When the value of x are small, the value of y are often easy to compute or estimate.**

**3) Graph the function at or near any end points of its domain.**