Review of Some Basic Graphs Needed for Calculus

Calculus professors believe that all of their students initially know a lot about the following functions. This belief is partly due to the fact that the professor cannot teach calculus to students that are not familiar with these functions. If you are a student of the calculus you should learn every detail covered in this worksheet. The graphs of all of these functions extend to infinity in at least one direction and it is part of the study of calculus to understand what these graphs do as they leave the small window in which we are graphing them. It is very important to know the range and domain of each of the following functions.

Linear Functions

Functions of the form have graphs that are straight lines. m is the slope and b is the y-intercept. If then the graph is a horizontal line.

The plot below is the graph of the function . The domain and the range for this function is the interval ( , ).

The plot below is the graph of the function . By reflecting the graph above across the x-axis we get the graph below. The domain and the range for this function is the interval ( , ).

Absolute Value Function

The plot below shows the absolute value function. Note that since f(-x) = | -x | = | x | = f(x) reflecting the absolute value function through the y-axis does not change the graph. The domain for this function is the interval ( , ). The range for f is the interval [ 0, ).

The plot below shows the absolute value function reflected through the x-axis and then translated two units to the right. Calculus students should know how to move the absolute value function by translating it up or down or reflecting through the x-axis. The range for g is the interval ( ,0].

Quadratic Functions

Functions of the form have graphs that are called parabolas if we assume that a is not zero. (If a is zero then the graph is a horizontal line.) If a quadratic function is writt en in the form , and we want to graph it, it is better to convert it to the form given as f above by completing the square.

The plot below is the standard parabola . The domain is the interval ( , ). The range for f is the interval [ 0, ).

The plot below is the curve . By shifting the graph of f up two units we get the graph of g. The range for g is the interval [ 2, ).

The Reciprocal of Function

The function is positive for all values in its domain. It has the line as a vertical asymptote and the line as a horizontal asymptote. The domain of this function consists of the union of the two intervals (- , 0) and (0 , ). The range is the one interval (0 , ). This function is not one-to-one so it is not invertible on its whole domain. This function is increasing for x in the interval (- , 0) and decreasing for x in the interval (0 , ). Note that is + . Also note that is + .

Reciprocal Function

The function is graphed below. It has the line as a vertical asymptote and the line as a horizontal asymptote. The domain and range of this function consists of the union of the two intervals (- , 0) and (0 , ). This function is one-to-one so it is invertible on its whole domain. In fact, this function is its own inverse. This function is decreasing for x in the interval (- , 0) and decreasing for x in the interval (0 , ). Note that is + . Also note that is - .

By rotating the coordinate system it is possible to put the equation into the standard form for the equation of a hyperbola.

Square Root Function

The graph of the square root function, , can be obtained by taking the part of the graph of that is in the first quadrant and flipping it across .The domain for f is the interval [0, ). The range for f is the interval [ 0, ). The limit as x goes to infinity for f(x) is infinity . The limit as x goes to zero from the right of f is zero. Note that both of these limits are a little difficult to see on the plot.

The graph of below, can be obtained by taking the part of the graph of that is in the second quadrant and flipping it across . The domain for g is the interval [0, ). The range for g is the interval ( ,0].

Cubic Functions

Graphing general cubics of the form where a is not zero is a difficult problem that you should be able to handle fairly well after you have finished Calculus I.

Beginning calculus students should know the following graphs of elementary cubics, and . Note that the graph of g is the graph of f translated up 3 units and 4 units to the left. The domain and the range for both graphs is the interval ( , ). Both graphs are increasing for all values in the domain. Both functions are one-to-one and therefore invertible and calculus students should be able to explicitly find the inverse finctions by solving for x.

Sine and Cosine Functions

Two periods of the sine function are graphed below. We need to be able to graph any function of the form . Note that 'A' gives the amplitude, 'a' represents a translation to the left or right, and b is a translation up or down. If 'A' is zero then the graph is just the horizontal line .

The following plot shows two periods of both the sine and cosine functions. Note that by shifting the sine function units to the left that we get the cosine function which means that . If we shift the sine function units to the right we can see by looking at the graph that we get the negative of the cosine function which means that . The domain for both functions is the interval ( , ). The range for both functions is the interval [ -1, 1].

Tangent Function

This graph has 4 periods of the tangent function, . To get the cotangent graph we only have to translate this graph units to the right and then reflect through the x-axis. The domain of the tangent function includes all real numbers except the odd multiples of .The range for the tangent function is the interval ( , ). The graph has vertical asymptotes at every odd multiple of so it actually has infinitely many vertical asymptotes . The limit as x goes to from the left is + . The limit as x goes to from the right is - . The tangent function is increasing on its whole domain.

Exponential Functions

Note that the exponential function, has a horizontal asymptote of y = 0. Also, note that when we translate the natural logarithm function two units down as we did in the second plot below that the asymptote has to move two units down to become y = -2. The domain for both graphs is the interval ( , ). The range for f is (0, ). The range for g is (-2 , ). Note that is 0 which verifies the horizontal asymptote for f.

Logarithm Functions

Note that the natural logarithm function has a vertical asymptote of x = 0. Also, note that when we translate the natural logarithm function two units to the left that the asymptote has to move two units to the left to become x = -2. The range for both graphs is the interval ( , ). The domain for f is (0, ). The domain for g is (-2 , ). Although it is difficult to deduce from the graphs, the limit of both of these functions as x goes to is which is a fact that is important in Calculus II.