Graphing Review Math Tutorial Lab Special Topic Common Functions and Their Graphs Linear Functions A function f defined b a linear equation of the form = f() = m + b, where m and b are constants, is called a linear function. Linear functions have graphs that are straight lines, and an nonvertical straight line is the graph of a linear function. We can compute the slope, m, of a line passing through (, ) and (, ), when, with the formula m =. If = (m = 0), the line is horizontal. Horizontal lines have the form = a for some constant a. If = (m is undefined), the line is vertical. Vertical lines have the form = c for some constant c. Warning: A vertical line is not a function! In our definition above, the equation = m + b is often called the slope-intercept form of a line, since m is the slope and b is the -intercept of the line. If a line passes through the point (, ), we compute b b b = m. In addition, a line with slope m that passes through the point (, ) has the point-slope equation = m( ). Lastl, linear equations are often written in their general form, A + B + C = 0, for constants A, B, and C. Eample A line passes through the points (, ) and (, 0). (a) Find a point-slope equation of this line. (b) Find the slope-intercept equation of this line. In the above eample, notice that reversing the order of the points when computing the slope will still give the same slope. Similarl, we could have used the other point to find the point-slope equation. The point-slope equation would look slightl different, but would still reduce to the same slope-intercept equation. Created b Maria Gommel, June 0.
Eample Sketch the graph of the linear function described b f() =. Quadratic Functions A function f defined b a quadratic equation of the form = f() = a + b + c, where a 0, is a quadratic function, and its graph is a parabola. The most basic parabola is defined b = f() =, whose graph is shown in Figure. The equation = f() = a + b + c is often called the standard form of the parabola. Quadratic equations can also be written in verte form: = f() = a( h) + k, where (h, k) is the verte of the parabola. The constant a stretches the graph verticall if a > and compresses the graph verticall if a <. Oftentimes, it is easier to graph a quadratic function in verte form. Converting from standard form to verte form usuall requires completing the square, as will be seen in the eample problems. Figure Figure Eample Sketch the graph of the quadratic equation = ( ) +.
Cubic Functions A function f defined b a cubic equation of the form = f() = a + b + c + d, where a 0, is a cubic function. The most basic cubic function is defined b = f() =, whose graph is shown in Figure. Square Root Function The graph of the square root function, f() =, is shown in Figure. Square roots are defined onl for nonnegative numbers, so f() = is defined onl for 0 and the domain of the square root function is the set of all nonnegative real numbers. 0 Figure Figure Piecewise-Defined Functions A function that is defined b differing epressions on various portions of its domain is called a piecewisedefined function. An eample of a piecewise-defined function is given b { f() =, if < 0, if, whose graph is seen in Figure. Graph Transformations Man times, we ma want to sketch a graph b hand. This is most often done b identifing a parent function, and then performing various graph transformations to obtain the graph we want. The various tpes of graph transformations are listed below, followed b a few eample problems. Horizontal Shifts Suppose that c > 0. The graph of = f( c) is the graph of = f() shifted right c units. The graph of = f( + c) is the graph of = f() shifted left c units.
Eample Compare the following graphs with the graph of. f() = ( ) f() = ( + ) Vertical Shifts Suppose c > 0. The graph of = f() + c is the graph of = f() shifted up c units. The graph of = f() c is the graph of = f() shifted down c units. Eample Compare the following graphs with the graph of. f() = 0 f() = + Reflecting over the aes The graph of = f() flips the graph of = f() over the ais. The graph of = f( ) flips the graph of = f() over the ais. Eample Compare the following graphs with the graph of. f() = f() =
Stretching or Compressing the graph The graph of = cf() stretches the graph of = f() verticall when c >, and compresses the graph verticall when 0 < c <. The graph of = f(c) compresses the graph of = f() horizontall when c > and stretches the graph horizontall when 0 < c <. Eample Compare the following graphs with the graph of. 0 0 0 0 0 0 0 0 0 0 f() = f() = f() = () f() = ( ) Eample We will give an eample of how to graph a rather complicated function using all the transformations above. Consider f() = +. Step : We first identif the parent function. Since we have a square root in our function, is our parent function. We graph first, and then transform the graph according to the rules above and the equation of f(). Step : We follow the order of operations. Think of the under the square root as an epression in parentheses: we ll perform this transformation first. The signals a horizontal shift to the right b one unit. We shift the graph of to the right b one. Step : Now, again following the order of operations, we ll consider the in front of the square root. Remember that the negative sign in this position flips the graph over the ais, and the stretches the graph verticall. Step : Lastl, we look at the + at the end of the equation. This is a vertical shift of units upwards. These steps are shown in the graphs below. Step : f() = Step : f() = Step : f() = Step : f() = +
Eample Problems Eample Sketch the graph of f() = +. (Hint: compare to the eample in the Quadratic Functions section). Eample Sketch the graph of f() = + + 7. Eample Sketch the graph of f() =. Eample Sketch the graph of f() =. Eample Sketch the graph of f() = {, if < 0, if 0. References Most definitions, eamples, and other tet was taken from the fifth edition of PreCalculus b J. Douglas Faires and James DeFranza.