## SAT Test Prep

## CHAPTER 11

ESSENTIAL ALGEBRA 2 SKILLS

### Lesson 2: Functions

**What Is a Function?**

A *function* is any set of instructions for turning an input number (usually called *x*) into an output number (usually called *y*). For instance, is a function that takes any input *x* and multiplies it by 3 and then adds 2. The result is the output, which we call *f (x*) or *y*.

If , what is *f* (2*h*)?

In the expression *f* (2*h*), the 2*h* represents the input to the function *f*. So just substitute 2*h* for *x* in the equation and simplify: .

**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 representations. 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 , where *m* is the slope of the line, and *b* is where the line intersects the *y*-axis. (For more on slopes, see __Chapter 10__, Lesson 4.)

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

Notice several important things about this table. First, as in every linear function, when the *x* values are “evenly spaced,” the *y* values are also “evenly spaced.” In this table, whenever the *x* value increases by 1, the *y* value increases by 3, which is the slope of the line and the coefficient of*x* in the equation. Notice 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:

**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 , where *c* is 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.”

The graph above represents the function . Notice that it is an “open down” parabola with an axis of symmetry through its vertex at .

The figure above shows the graph of the function *f* in the *xy*-plane. If , 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 . (You can plug into the equation to verify.) Since this point is two units from the axis of symmetry, its reflection is two units on the *other* side of the axis, which is the point (4, –3).

**Concept Review 2: Functions**

__1.__ What is a function?

__2.__ What are the three basic ways of representing a function?

__3.__ What is the general form of the equation of a linear function, and what does the equation tell you about the graph?

__4.__ How can you determine the slope of a linear function from a table of its inputs and outputs?

__5.__ How can you determine the slope of a linear function from its graph?

__6.__ What is the general form of the equation of a quadratic function?

__7.__ What kind of symmetry does the graph of a quadratic function have?

**SAT Practice 2: Functions**

**1**__.__ The graphs of functions *f* and *g* for values of *x* between –3 and 3 are shown above. Which of the following describes the set of all *x* for which

or

**2**__.__ If and , which of the following could be *g (x*)?

(A) 3 *x*

**3**__.__ What is the least possible value of if

(A) –3

(B) –2

(C) –1

(D) 0

(E) 1

**4**__.__ The table above gives the value of the linear function *f* for several values of *x*. What is the value of

(A) 8

(B) 12

(C) 16

(D) 24

(E) It cannot be determined from the information given.

**5**__.__ The graph on the *xy*-plane of the quadratic function *g* is a parabola with vertex at (3, –2). If , then which of the following must also equal 0?

(A) *g* (2)

(B) *g* (3)

(C) *g* (4)

(D) *g* (6)

(E) *g* (7)

**6**__.__ In the *xy*-plane, the graph of the function *h* is a line. If and , what is the value of *h* (0)?

(A) 2.0

(B) 2.2

(C) 3.3

(D) 3.5

(E) 3.7

**Answer Key 2: Functions**

**Concept Review 2**

__1.__ A set of instructions for turning an input number (usually called *x*) into an output number (usually called *y*).

__2.__ As an equation (as in ), as a table of input and output values, and as a graph in the *xy*-plane.

, where *m* is the slope of the line and *b* is its *y*-intercept.

__4.__ If the table provides two ordered pairs, (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), the slope can be calculated with . (Also see __Chapter 10__, Lesson 4.)

__5.__ Choose any two points on the graph and call their coordinates (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}). Then calculate the slope with .

, where *c* is the *y*-intercept.

__7.__ It is a parabola that has a vertical line of symmetry through its vertex.

**SAT Practice 2**

__1.__ **C** In this graph, saying that is the same as saying that the *g* function “meets or is above” the *f* function. This is true between the points where they meet, at and .

__2.__ **B** Since , *f (g* (1)) must equal . Therefore and . So *g (x*) must be a function that gives an output of 4 when its input is 1. The only expression among the choices that equals 4 when is .

__3.__ **D** This question asks you to analyze the “outputs” to the function given a set of “inputs.” Don”t just assume that the least input, –3, gives the least output, . In fact, that”s not the least output. Just think about the arithmetic: is the square of a number. What is the least possible square of a real number? It must be 0, because 0^{2} equals 0, but the square of any other real number is positive. Can in this problem equal 0? Certainly, if , which is in fact one of the allowed values of *x*. Another way to solve the problem is to notice that the function is quadratic, so its graph is a parabola. Choose values of *x* between –3 and 0 to make a quick sketch of this function to see that its vertex is at (–2, 0).

__4.__ **C** Since *f* is a linear function, it has the form . The table shows that an input of 3 gives an output of 8, so . Now, if you want, you can just “guess and check” values for *m* and *b* that work, for instance, and . This gives the equation . To find the missing outputs in the table, just substitute and then : and . Therefore, . But how do we know that will __always__ equal 16? Because the slope *m* of any linear function represents the amount that*y* increases (or decreases) whenever *x* increases by 1. Since the table shows *x* values that increase by 1, *a* must equal , and *b* must equal 8 + *m*. Therefore .

__5.__ **D** Don”t worry about actually finding the equation for *g (x*). Since *g* is a quadratic function, it has a vertical line of symmetry through its vertex, the line . Since , the graph also passes through the origin. Draw a quick sketch of a parabola that passes through the origin and (3, –2) and has an axis of symmetry at :

The graph shows that the point (0, 0), when reflected over the line , gives the point (6, 0). Therefore *g* (6) is also equal to 0.

__6.__ **D** The problem provides two ordered pairs that lie on the line: (–1, 4) and (5, 1). Therefore, the slope of this line is . Therefore, for every one step that the line takes to the right (the *x* direction), the *y* value *decreases* by ½. Since 0 is one unit to the right of –1 on the *x*-axis, *h* (0) must be^{1}/_{2} less than *h* ((–1), or .