Mastering Nonlinear Functions for the SAT

Introduction

Welcome to this comprehensive lesson on nonlinear functions for the SAT! Nonlinear functions are a crucial concept that appears frequently on the SAT Math section. Understanding these functions will not only help you solve specific problems but also develop a deeper mathematical intuition that will benefit you across various question types. This lesson is designed for students with no prior knowledge of nonlinear functions, and we'll build your understanding from the ground up.

What are Nonlinear Functions?

Nonlinear functions are mathematical relationships between variables that don't form a straight line when graphed. Unlike linear functions (which follow the form $y = m x + b$ and create straight lines), nonlinear functions create curves, circles, parabolas, or other non-straight shapes when plotted.

The most common types of nonlinear functions you'll encounter on the SAT include:

Quadratic Functions: These have the form $y = a x^{2} + b x + c$ where $a \neq = 0$ . They create parabolas when graphed.
Exponential Functions: These have the form $y = a \cdot b^{x}$ where $a$ and $b$ are constants and $b > 0, b \neq = 1$ . They show rapid growth or decay.
Absolute Value Functions: These contain absolute value expressions like $y = ∣ x - 3∣$ and create V-shaped graphs.
Rational Functions: These involve fractions with variables in the denominator, like $y = \frac{1}{x}$ or $y = \frac{x + 1}{x - 2}$ .
Radical Functions: These involve square roots or other roots, like $y = x$ or $y = 3 x + 2$ .

Nonlinear functions are essential for modeling real-world phenomena that don't change at a constant rate, such as population growth, compound interest, or the trajectory of a thrown ball.

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How to Use Nonlinear Functions

To effectively work with nonlinear functions on the SAT, follow these key strategies:

1. Identifying Nonlinear Functions

Look for terms with exponents (like $x^{2}$ , $x^{3}$ ), square roots, absolute value symbols, or variables in denominators.
Remember that any function that isn't in the form $y = m x + b$ is nonlinear.

2. Graphing Nonlinear Functions

For quadratics ( $y = a x^{2} + b x + c$ ):
- The parabola opens upward if $a > 0$ and downward if $a < 0$ .
- The vertex formula is $x = - \frac{b}{2 a}$ .
- Substitute this x-value back into the function to find the y-coordinate of the vertex.
For exponential functions ( $y = a \cdot b^{x}$ ):
- If $b > 1$ , the function grows exponentially.
- If $0 < b < 1$ , the function decays exponentially.
- The y-intercept is always at $(0, a)$ .

3. Finding Zeros/Roots

For quadratics, use the quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ .
Alternatively, factor when possible: $a x^{2} + b x + c = 0$ might factor into $(x - r) (x - s) = 0$ , giving roots at $x = r$ and $x = s$ .

4. Transformations

Vertical shifts: $f (x) + k$ shifts the graph up by $k$ units.
Horizontal shifts: $f (x - h)$ shifts the graph right by $h$ units.
Reflections: $- f (x)$ reflects across the x-axis; $f (- x)$ reflects across the y-axis.
Stretches/compressions: $a \cdot f (x)$ stretches vertically by a factor of $∣ a ∣$ .

5. Solving Nonlinear Equations

Set the function equal to the desired value and solve for the variable.
For quadratics, rearrange to standard form and use factoring or the quadratic formula.
For other types, isolate the variable using appropriate algebraic techniques.

6. Analyzing Key Features

Domain and range: Determine what x and y values are valid for the function.
Increasing/decreasing intervals: Identify where the function goes up or down.
Maximum/minimum values: Find the highest or lowest points on the graph.
End behavior: Determine what happens to y-values as x approaches positive or negative infinity.

Nonlinear Functions Worksheet

Part A: Identifying Nonlinear Functions

Classify each of the following as linear or nonlinear. If nonlinear, specify the type (quadratic, exponential, etc.).

$y = 3 x + 7$
$y = x^{2} - 4$
$y = 2^{x}$
$y = \frac{5}{x + 1}$
$y = ∣ x - 3∣$

Part B: Finding Key Points

For each quadratic function, find the vertex, y-intercept, and x-intercepts (if they exist).

$y = x^{2} - 6 x + 8$
$y = - 2 x^{2} + 4 x - 1$

Part C: Graphing

Sketch the graphs of the following functions, showing key points:

$y = x^{2} - 4$
$y = 2^{x}$
$y = \frac{1}{x}$

Part D: Solving Nonlinear Equations

Solve for x in each equation:

$x^{2} - 5 x + 6 = 0$
$2^{x} = 8$
$x + 3 = 4$

Part E: Applications

A ball is thrown upward from a height of 6 feet with an initial velocity of 32 feet per second. The height $h$ of the ball after $t$ seconds is given by the function $h = - 16 t^{2} + 32 t + 6$ .

What is the maximum height reached by the ball?
At what time does the ball reach its maximum height?
When does the ball hit the ground?

Answers are provided at the end of this worksheet for self-checking.

Nonlinear Functions Examples

Example 1

Quadratic Function Example

Question: Find the vertex and x-intercepts of $f (x) = x^{2} - 6 x + 5$ .

Solution:

To find the vertex, use the formula $x = - \frac{b}{2 a}$ :
$x = - \frac{- 6}{2 ( 1 )} = \frac{6}{2} = 3$
Find the y-coordinate by substituting $x = 3$ into the function:
$f (3) = 3^{2} - 6 (3) + 5 = 9 - 18 + 5 = - 4$
The vertex is at the point $(3, - 4)$ .
For x-intercepts, set $f (x) = 0$ and solve:
$x^{2} - 6 x + 5 = 0$
$(x - 5) (x - 1) = 0$
$x = 5$ or $x = 1$
The x-intercepts are at $x = 1$ and $x = 5$ .

Example 2

Exponential Function Example

Question: Solve the equation $2^{x} = 32$ .

Solution:

Recognize that $32 = 2^{5}$ .
So the equation becomes $2^{x} = 2^{5}$ .
Since the bases are the same, the exponents must be equal: $x = 5$ .

Alternatively, using logarithms:

Take $lo g_{2}$ of both sides: $lo g_{2} (2^{x}) = lo g_{2} (32)$ .
Simplify: $x = lo g_{2} (32)$ .
Since $32 = 2^{5}$ , we have $x = 5$ .

Example 3

Absolute Value Function Example

Question: Solve $∣2 x - 6∣ = 10$ .

Solution:

The absolute value equals 10, so there are two possibilities:
- $2 x - 6 = 10$ or $2 x - 6 = - 10$
Solve the first equation:
$2 x - 6 = 10$
$2 x = 16$
$x = 8$
Solve the second equation:
$2 x - 6 = - 10$
$2 x = - 4$
$x = - 2$
The solutions are $x = 8$ and $x = - 2$ .

Example 4

Rational Function Example

Question: Find the domain of $f (x) = \frac{x + 3}{x ^{2} - 4}$ .

Solution:

A rational function is undefined when its denominator equals zero.
Set the denominator equal to zero and solve:
$x^{2} - 4 = 0$
$(x + 2) (x - 2) = 0$
$x = - 2$ or $x = 2$
The domain is all real numbers except $x = - 2$ and $x = 2$ .
In set notation: Domain = ${x ∣ x \in R, x \neq = - 2, x \neq = 2}$ .

Example 5

Radical Function Example

Question: Find the domain and range of $f (x) = x - 4$ .

Solution:

For the domain, the expression under the square root must be non-negative:
$x - 4 \geq 0$
$x \geq 4$
The domain is ${x ∣ x \geq 4}$ or $[4, \infty)$ .
For the range, since the square root function always produces non-negative values, and we can get arbitrarily large values by inputting large numbers for $x$ , the range is ${y ∣ y \geq 0}$ or $[0, \infty)$ .

Example 6

SAT-Style Application Example

Question: A projectile is launched from ground level with an initial velocity of 80 feet per second. Its height $h$ in feet after $t$ seconds is given by the function $h = - 16 t^{2} + 80 t$ . What is the maximum height reached by the projectile?

Solution:

To find the maximum height, we need to find the vertex of this parabola.
Using the vertex formula $t = - \frac{b}{2 a}$ :
$t = - \frac{80}{2 ( - 16 )} = \frac{80}{32} = 2.5$ seconds
Substitute this value of $t$ into the height function:
$h = - 16 (2.5)^{2} + 80 (2.5)$
$h = - 16 (6.25) + 80 (2.5)$
$h = - 100 + 200$
$h = 100$ feet
The maximum height reached by the projectile is 100 feet.

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Common Misconceptions

When working with nonlinear functions, students often encounter these misconceptions:

1. Confusing Linear and Nonlinear Functions

Misconception: Any equation with $x$ and $y$ is linear.
Reality: Linear functions must have the form $y = m x + b$ with no higher powers of variables, no products of variables, and no variables in denominators or under radicals.

2. Vertex Confusion in Quadratics

Misconception: The vertex of $y = a x^{2} + b x + c$ is at $x = b /2 a$ .
Reality: The x-coordinate of the vertex is $x = - b /2 a$ (note the negative sign).

3. Exponential vs. Quadratic Growth

Misconception: $x^{2}$ and $2^{x}$ grow at similar rates.
Reality: Exponential functions like $2^{x}$ eventually grow much faster than any polynomial function like $x^{2}$ .

4. Domain Oversights

Misconception: All functions have the same domain (all real numbers).
Reality: Many nonlinear functions have restricted domains. For example, square root functions require the expression under the radical to be non-negative, and rational functions exclude values that make the denominator zero.

5. Solving Absolute Value Equations

Misconception: $∣ x ∣ = 5$ means $x = 5$ .
Reality: Absolute value equations typically have two solutions. $∣ x ∣ = 5$ means $x = 5$ OR $x = - 5$ .

6. Graphing Confusion

Misconception: The graph of $y = x^{2} + 2$ is the same as $y = (x + 2)^{2}$ .
Reality: $y = x^{2} + 2$ shifts the parabola up by 2 units, while $y = (x + 2)^{2}$ shifts it left by 2 units and changes its shape.

7. Factoring Limitations

Misconception: All quadratic equations can be factored easily.
Reality: Many quadratics don't factor nicely with integer coefficients. The quadratic formula works for all quadratics.

8. Misinterpreting Negative Exponents

Misconception: $2^{- 3}$ is negative.
Reality: $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ , which is positive but less than 1.

Practice Questions for Nonlinear Functions

Question 1

The function $h (t) = - 16 t^{2} + 64 t + 6$ models the height in feet of a ball $t$ seconds after it is thrown. What is the maximum height reached by the ball?

A) 6 feet
B) 64 feet
C) 70 feet
D) 86 feet

Solution:
To find the maximum height, we need to find the vertex of this parabola.

Step 1: Find the t-value at the vertex using $t = - \frac{b}{2 a}$ .
$t = - \frac{64}{2 ( - 16 )} = \frac{64}{32} = 2$ seconds

Step 2: Substitute this t-value into the original function.
$h (2) = - 16 (2)^{2} + 64 (2) + 6$
$h (2) = - 16 (4) + 128 + 6$
$h (2) = - 64 + 128 + 6$
$h (2) = 70$ feet

The maximum height reached by the ball is 70 feet.

The answer is C) 70 feet.

Question 2

If $f (x) = 3^{x}$ and $g (x) = 3^{x + 2}$ , which of the following expresses $g (x)$ in terms of $f (x)$ ?

A) $g (x) = f (x) + 2$
B) $g (x) = f (x + 2)$
C) $g (x) = f (x) \cdot 3^{2}$
D) $g (x) = 2 f (x)$

Solution:
We need to express $g (x) = 3^{x + 2}$ in terms of $f (x) = 3^{x}$ .

Step 1: Rewrite $g (x)$ using properties of exponents.
$g (x) = 3^{x + 2} = 3^{x} \cdot 3^{2} = 3^{x} \cdot 9$

Step 2: Substitute $f (x) = 3^{x}$ .
$g (x) = f (x) \cdot 9$

However, this exact option isn't listed. Let's check the closest one:
$g (x) = f (x) \cdot 3^{2} = f (x) \cdot 9$

The answer is C) $g (x) = f (x) \cdot 3^{2}$ .

Nonlinear Functions Questions

Question 1

The function $f (x) = a x^{2} + b x + c$ has its vertex at the point (3, -2) and passes through the point (0, 7). What is the value of $a$ ?

A) -1
B) 1
C) 2
D) 3

Solution:
Step 1: Since the vertex is at (3, -2), we can rewrite the function in vertex form:
$f (x) = a (x - 3)^{2} - 2$

Step 2: Use the point (0, 7) to find the value of $a$ :
$f (0) = a (0 - 3)^{2} - 2 = 7$
$a (9) - 2 = 7$
$9 a = 9$
$a = 1$

The answer is B) 1.

Question 2

If $f (x) = ∣ x - 4∣ + ∣ x + 2∣$ , what is the minimum value of $f (x)$ ?

A) 0
B) 2
C) 6
D) 8

Solution:
Step 1: Recognize that the minimum value of $∣ x - 4∣ + ∣ x + 2∣$ occurs at a point between -2 and 4.

Step 2: For any point $x$ in this interval, we have:
$∣ x - 4∣ = 4 - x$ (since $x < 4$ )
$∣ x + 2∣ = x + 2$ (since $x > - 2$ )

So $f (x) = (4 - x) + (x + 2) = 6$ for any $x$ in the interval $[- 2, 4]$ .

Step 3: Check values outside this interval to confirm they give larger results.

The minimum value of $f (x)$ is 6.

The answer is C) 6.

Nonlinear Functions Learning Checklist

I can identify different types of nonlinear functions (quadratic, exponential, absolute value, rational, radical).
I can find the vertex, axis of symmetry, and intercepts of a quadratic function.
I can solve quadratic equations using factoring, completing the square, and the quadratic formula.
I can graph basic nonlinear functions and identify their key features.
I can solve exponential equations using properties of exponents and logarithms.
I can determine the domain and range of nonlinear functions.
I can solve absolute value equations and inequalities.
I can apply transformations to nonlinear functions (shifts, stretches, reflections).
I can solve word problems involving nonlinear functions, particularly quadratic models.
I can distinguish between different types of growth (linear, quadratic, exponential) in real-world contexts.

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