Graphical Representations 

A function f has a graph in the xy-plane, which is the graph of the equation y = f (x), or, equivalently, consists of all ordered pairs (x, f (x)). Some Passport to Advanced Math questions assess your ability to relate properties of the function f to properties of its graph, and vice versa. You may be required to apply some of the following relationships:

• Intercepts: The x-intercepts of the graph of f correspond to values of x such that f (x) = 0, which corresponds to where the graph intersects with the x-axis; if the function f has no zeros, its graph has no x-intercepts, and vice versa. The y-intercept of the graph of f corresponds to the value of f (0), or where the graph intersects with the y-axis. If x = 0 is not in the domain of f, the graph of f has no y-intercept, and vice versa.

• Domain and range: The domain of f is the set of all x for which f (x) is defined. The range of f is the set of all y such that y = f (x) for some value of x in the domain. The domain and range can be found from the graph of f as the set of all x-coordinates and y-coordinates, respectively, of points on the graph.

• Maximum and minimum values: The maximum and minimum values of f can be found by locating the highest and the lowest points on the graph, respectively. For example, suppose P is the highest point on the graph of f. Then the y-coordinate of P is the maximum value of f, and the x-coordinate of P is where f takes on its maximum value.

• Increasing and decreasing: The graph of f shows the intervals over which the function f is increasing and decreasing end behavior. The graph of f can indicate if f (x) increases or decreases without limit as x increases or decreases without limit. 

• Transformations: For a graph of a function f, a change of the form f (x) + a will result in a vertical shift of a unit, and a change of the form f (x + a) will result in a horizontal shift of a units. 

The SAT Math Test uses the following conventions about graphs in the xy-plane unless a particular question clearly states or shows a different convention: 

• The axes are perpendicular. 

• Scales on the axes are linear scales. 

• The size of the units on the two axes cannot be assumed to be equal unless the question states they are equal or you are given enough information to conclude they are equal. 

• The values on the horizontal axis increase as you move to the right. 

• The values on the vertical axis increase as you move up.

The domain of a function is the set of all values for which the function is defined. The range of a function is the set of all values that correspond to the values in the domain, given the relationship defined by the function, or the set of all outputs that are associated with all of the possible inputs.