SAT Mc Graw Hill 2011

412 McGRAW-HILL’S SAT

Lesson 2: Functions

What Is a Function?

A functionis any set of instructions for turning

an input number (usually called x) into an output

number (usually called y). For instance, f(x) =

3 x+ 2 is a function that takes any input xand

multiplies it by 3 and then adds 2. The result is

the output, which we call f(x) or y.

If f(x) = 3x+ 2, what is f(2h)?

In the expression f(2h), the 2hrepresents the input
to the function f. So just substitute 2hfor xin the
equation and simplify: f(2h) = 3(2h) + 2 = 6h+2.

Functions as Equations, Tables, or Graphs

The SAT usually represents a function in one of

three ways: as an equation, as a table of inputs

and outputs, or as a graph on the xy-plane. Make

sure that you can work with all three represen-

tations. For instance, know how to use a table to

verify an equation or a graph, or how to use an

equation to create or verify a graph.

Linear Functions

A linear function is any function whose graph is a

line. The equations of linear functions always have

the form f(x) = mx+b,where mis the slope of the

line, and bis where the line intersects the y-axis.

(For more on slopes, see Chapter 10, Lesson 4.)

The function f(x) = 3x+ 2 is linear with a slope of 3
and a y-intercept of 2. It can also be represented with a
table of xand y(or f(x)) values that work in the equation:

also that the y-intercept is the output to the function

when the input is 0.

Now we can take this table of values and plot each

ordered pair as a point on the xy-plane, and the result

is the graph of a line:

x f(x)

–2

2

3

4

1

–1

0

–4

5

8

11

14

–1

2

y

x

1

f(x) = 3x + 2

1

Quadratic Functions

The graph of a quadratic function is always a

parabola with a vertical axis of symmetry. The

equations of quadratic functions always have

the form f(x) = ax^2 + bx+c,where cis the

y-intercept. When a(the coefficient of x^2 ) is

positive, the parabola is “open up,” and when a

is negative, it is “open down.”

y

x

1

f(x) = –x^2 + 4x – 3

1

The graph above represents the function y= x^2 +

4 x3. Notice that it is an “open down” parabola with

an axis of symmetry through its vertex at x= 2.

The figure above shows the graph of the function

fin the xy-plane. If f(0) = f(b), which of the following

could be the value of b?

(A)3(B)2 (C) 2 (D) 3 (E) 4

Although this can be solved algebraically, you

should be able to solve this problem more simply just

by inspecting the graph, which clearly shows that

f(0) = 3. (You can plug x= 0 into the equation to

verify.) Since this point is two units from the axis of

symmetry, its reflection is two units on the otherside

of the axis, which is the point (4, 3).

Notice several important things about this table.
First, as in every linear function, when the xvalues
are “evenly spaced,” the yvalues are also “evenly
spaced.” In this table, whenever the xvalue increases
by 1, the yvalue increases by 3, which is the slope of
the line and the coefficient of xin the equation. Notice