Nonlinear functions

Explanation

Nonlinear Functions

Nonlinear functions represent relationships where the rate of change between variables is not constant. On the SAT, these primarily include quadratic functions, exponential functions, and basic polynomial or rational functions.

Quadratic Functions

Quadratic functions form parabolas and can be represented in three primary forms:

Standard Form: $f(x) = ax^2 + bx + c$
- $y$ -intercept is $(0, c)$ .
- Axis of symmetry is $x = -\dfrac{b}{2a}$ .
- Vertex $x$ -coordinate is $x = -\dfrac{b}{2a}$ .
Vertex Form: $f(x) = a(x - h)^2 + k$
- Vertex is directly given as $(h, k)$ .
- If $a > 0$ , the parabola opens upward (minimum value is $k$ ).
- If $a < 0$ , the parabola opens downward (maximum value is $k$ ).
Factored (Intercept) Form: $f(x) = a(x - r_1)(x - r_2)$
- $x$ -intercepts (roots/zeros) occur at $(r_1, 0)$ and $(r_2, 0)$ .
- Axis of symmetry lies midpoint between roots: $x = \dfrac{r_1 + r_2}{2}$ .

Exponential Functions

Exponential functions model growth or decay by a constant percentage or multiplicative factor over equal intervals.

f(x) = a \cdot b^x \quad \text{or} \quad f(t) = P(1 \pm r)^t

Initial Value ( $a$ or $P$ ): The value when input is $0$ ( $y$ -intercept).
Growth Factor ( $b > 1$ ): Represented as $1 + r$ , where $r$ is the growth rate as a decimal.
Decay Factor ( $0 < b < 1$ ): Represented as $1 - r$ , where $r$ is the decay rate as a decimal.

Example 1

The function

f

is defined by

f(x)=(x-2)(x-10)

. For what value of

x

does

f

reach its minimum?

2

6

10

-16

Choice B is correct. Since

f(x)=(x-2)(x-10)

is a parabola with a positive leading coefficient, it opens upward and reaches its minimum at its vertex, which lies exactly midway between the two roots,

x=2

and

x=10

. The midpoint is

\frac{2+10}{2}=6

. Choice A is incorrect; this is one of the

x

-intercepts, not the vertex. Choice C is incorrect; this is the other

x

-intercept, not the vertex. Choice D is incorrect; this is the minimum value of

f

(that is,

f(6)=(6-2)(6-10)=-16

), not the value of

x

at which it occurs.

Example 2

The function

f

is defined by

f(x)=-a^x+b

, where

a

and

b

are constants. In the

xy

-plane, the graph of

f

has a

y

-intercept at

(0,-9)

. The product of

a

and

b

-40

. What is the value of

b

-8

Correct

Since

f(0)=-a^0+b=-1+b

for any nonzero constant

a

, and the

y

-intercept is

(0,-9)

, it follows that

-1+b=-9

, so

b=-8

. As a check, since the product of

a

and

b

-40

a=\frac{-40}{b}=\frac{-40}{-8}=5

, a valid positive base. Therefore,

b=-8

Tips for solving

Switch to Vertex Form to Find Maximum/Minimum. When an SAT question asks for the maximum or minimum value of a quadratic function, identify the vertex $(h, k)$ . The extremum is always the $y$ -value ( $k$ ).
Example: What is the minimum value of the function $f(x) = x^2 - 8x + 19$ ?
- Find the $x$ -coordinate of the vertex ( $h = -\frac{b}{2a}$ ):
  $h = -\frac{-8}{2(1)} = 4$
- Evaluate $f(4)$ to find the minimum value ( $k$ ):
  $f(4) = (4)^2 - 8(4) + 19 = 16 - 32 + 19 = 3$
- The minimum value of the function is $3$ .
Identify Exponential Growth/Decay Rates Instantly Be careful to distinguish between the growth factor ( $b$ ) and the growth rate ( $r$ ).
Example: The population of a culture of bacteria is modeled by $P(t) = 250(1.14)^t$ , where $t$ is measured in hours. By what percentage does the population grow each hour?
- Express the base in $(1 + r)$ form:
  $1.14 = 1 + 0.14$
- Convert the decimal rate to a percentage:
  $r = 0.14 \implies 14\%$
- The population grows by $14\%$ each hour.
Use Symmetry to Find the Vertex from Roots. If a quadratic function's zeros are given, the vertex always lies directly on the vertical line of symmetry halfway between them.
Example: A quadratic function $g(x)$ has zeros at $x = -2$ and $x = 8$ . For what value of $x$ does $g(x)$ reach its maximum or minimum?
- Calculate the midpoint of the roots:
  $x = \frac{-2 + 8}{2} = \frac{6}{2} = 3$
- The function reaches its extremum at $x = 3$ .

Practice questions

00:00

Question 1 · easy

A savings account balance, in dollars, after

t

years is given by

B(t)=2500(1.04)^t

. What does the value

2500

represent in this context?

Question 2 · easy

A population of bacteria is modeled by

P(t)=800(0.85)^t

, where

t

is time in hours. What does the value

0.85

indicate about the population?

Question 3 · easy

A car's value, in dollars, is modeled by

V(t)=32{,}000(0.92)^t

, where

t

is the number of years since the car was purchased. What is the initial value of the car, in dollars?

Question 4 · easy

The function

f(x)=1200(1.06)^x

models the value of an investment, in dollars, after

x

years. What is the growth factor per year?

Question 5 · easy

A geometric sequence has a first term of

9

and a common ratio of

4

. What is the fourth term of the sequence?

Question 6 · easy

A geometric sequence has first term

a_1=6

and common ratio

r=3

. Which of the following gives the

n

th term,

a_n

, of the sequence?

Question 7 · easy

The first term of a geometric sequence is

5

, and each subsequent term is

2

times the previous term. What is the fifth term of the sequence?

Question 8 · easy

A geometric sequence is defined by

w_n=7(2)^{n-1}

. What is the first term of the sequence,

w_1

Question 9 · easy

The function

f

is defined by

f(x)=(x-3)(x+5)

. If

g(x)=f(x)+10

, what is the value of

g(3)

Question 10 · easy

The function

f

is defined by

f(x)=(x-2)(x-6)

. If

g(x)=f(x)-4

, what is the value of

g(2)

SAT Prep

Nonlinear functions

Explanation