EXERCISE 3WORK OUT EACH PROBLEM. CIRCLE THE LETTER THAT APPEARS BEFORE...

4. GRAPHS OF FUNCTIONS AND OTHER

EQUATIONS—FEATURES AND TRANSFORMATIONS

On the new SAT, a question might show a graph of a quadratic function or other equation in the xy-plane, and thenask you to identify or recognize certain features of the graph—for example, minimum or maximum points on thegraph. You might encounter the graph of a circle, an ellipse, a parabola, or even a trigonometric function (appearingas a wave). To answer these questions, you do not need to know the equations that define such graphs; simply applyyour knowledge of the xy-coordinate system and, for some questions, function notation (see Chapter 15).Example:The figure above shows the graph of a certain equation in the xy-plane. The graph is a circle withcenter O and circumference 6π. At how many different values of y does x = –7.5 ?(A) 0(B) 1(C) 2(D) 4(E) Infinitely manySolution:The correct answer is (C). First, find the circle’s radius from its circumference: C = 6π = 2πr; r = 3.Since the circle’s center (0) lies at (–5,–6), the minimum value in the domain of x is –8. In otherwords, the left-most point along the circle’s circumference is at (–8,–6), 3 units to the left of O.Thus, the graph of x = –7.5, which is a vertical line passing through (–7.5, 0), intersects the circle atexactly two points. That is, when x = –7.5, there are two different corresponding values of y.Other questions on the new SAT will involve transformations of linear and quadratic functions and the effectof transformations on the graphs of such functions. The function f(x) is transformed by substituting an expressioncontaining the variable x for x in the function — for example:If f(x) = 2x, then f(x + 1) = 2(x + 1), or 2x + 2Transforming a function alters the graph of the function in the xy-plane. The effect of a transformation mightbe any of the following:* To move, or translate, the graph (either vertically, horizontally, or both) to another position in the plane* To alter the slope of a line (in the case of a linear function)* To alter the shape of a curve (in the case of a quadratic function)For example, if f(x) = x, then f(x + 1) = x + 1. In the xy-plane, the graph of f(x) = x (or y = x), is a line with slope1 passing through the origin (0,0). The effect of transforming f(x) to f(x + 1) on the graph of f(x) is the translationof the line one unit upward. (The y-intercept becomes 1 instead of 0.) Remember: In determining the graph of afunction in the xy-plane, use y to signify f(x) and, conversely, use x to signify f(y).If f(x) = x + 3, then the line shown in the xy-plane above is the graph of(A) f(x)(B) f(x – 6)(C) f(x + 6)(D) f(x + 3)(E) f(x – 3)The correct answer is (E). The figure shows the graph of the function f(x) = x (or y = x). Todetermine which of the five answer choices transforms the original function f(x) = x + 3 to thefunction f(x) = x, substitute the variable expression in each choice, in turn, for x in the originalfunction. Choice (E) is the only one that provides an expression that achieves this transformation:

−

=

−

+

f x

x

(

)

(

)

3

3

3

−

=

(

)

3

=

y

x

To help you determine the effect of a function’s transformation on the function’s graph, you can tabulate some(x,y) pairs based on the new function, plot the points on the xy-plane, and then connect them.If f(x) = x

²

, then the graph shown in the xy-plane above best represents which of the followingfunctions?(A) f(–x)(B) f(x – 1)(C) f(x + 1)(D) f(x

²

+ 1)(E) f(x

²

– 1)The correct answer is (B). The figure shows the graph of y = x

²

, but translated to the right.Substitute the variable expression given in each answer choice, in turn, for x in the function f(x) =x

²

. Performing this task for choice (B) yields the equation f(x) = (x – 1)

²

, or y = (x – 1)

²

. Identify andplot some (x,y) pairs. (Since the vertex in the graph lies along the x-axis, let x = 0 in order toestablish the vertex’s coordinates.) Here are some (x,y) pairs for the equation y = (x – 1)

²

:(0,1)(1,0), (2,1), (3,4), (–1,4)Plotting these points in the xy-plane reveals a graph whose key features match those of the figureprovided in the question.