will always equal 16? Because the slope mof any
linear function represents the amount that yin-
creases (or decreases) whenever xincreases by 1.
Since the table shows xvalues that increase by 1, a
must equal 8 −m, and bmust equal 8 +m. There-
fore a+b=(8 −m) +(8 +m)=16.
5.D Don’t worry about actually finding the equation
for g(x). Since gis a quadratic function, it has a
vertical line of symmetry through its vertex, the line
x=3. Since g(0) =0, 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 x=3:Concept Review 2
- A set of instructions for turning an input number
(usually called x) into an output number (usually
called y). - As an equation (as in f(x) = 2 x), as a table of input
and output values, and as a graph in the xy-plane.
3.f(x) =mx+b,where mis the slope of the line and
bis its y-intercept.
- 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.)
yy
xx
21
21−
−
- 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.6.f(x) =ax^2 +bx+c,where cis the y-intercept.- It is a parabola that has a vertical line of symmetry
through its vertex.
yy
xx21
21−
−
Answer Key 2: Functions
SAT Practice 2
1.C In this graph, saying that g(x) ≥f(x) is the same
as saying that the gfunction “meets or is above”
the ffunction. This is true between the points
where they meet, at x=−1 and x=2.
2.B Since f(x) =x+2, f(g(1)) must equal g(1) +2.
Therefore g(1) + 2 =6 and g(1) =4. 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 x=1 is (B) x+3.
3.D This question asks you to analyze the “outputs”
to the function y=(x+2)^2 given a set of “inputs.”
Don’t just assume that the least input, −3, gives the
least output, (− 3 +2)^2 =1. In fact, that’s not the
least output. Just think about the arithmetic:
(x+2)^2 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 x+2 in this problem
equal 0? Certainly, if x=−2, which is in fact one of
the allowed values of x.Another way to solve the
problem is to notice that the function y=(x+2)^2 is
quadratic, so its graph is a parabola. Choose values
of xbetween −3 and 0 to make a quick sketch of this
function to see that its vertex is at (−2, 0).
4.C Since fis a linear function, it has the form f(x) =
mx+b. The table shows that an input of 3 gives an
output of 8, so 3m+b=8. Now, if you want, you
can just “guess and check” values for mand bthat
work, for instance, m=2 and b=2. This gives the
equation f(x) = 2 x+2. To find the missing outputs
in the table, just substitute x=2 and then x=4:
f(2) =2(2) + 2 =6 and f(4) =2(4) + 2 =10. Therefore,
a+b= 6 + 10 =16. But how do we know that a+b
CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 415
yxy = g(x)OThe graph shows that the point (0, 0), when
reflected over the line x=3, 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 (4 −1)/(− 1 −5) =−3/6 =−1/2. Therefore,
for every one step that the line takes to the right
(the xdirection), the yvalue decreasesby^1 / 2. 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 4 −1/2 =3.5.