Nonlinear Functions – SAT Math

📖 What Are Nonlinear Functions?

A nonlinear function is a function whose graph is not a straight line. The rate of change is not constant.

Quadratic Function

$y = ax^2 + bx + c$

Parabola shape. $a > 0$ → opens up, $a < 0$ → opens down

Exponential Function

$y = a \cdot b^x$

Rapid growth or decay. $b > 1$ → growth, $0 < b < 1$ → decay

Polynomial Function

$y = a_n x^n + ... + a_1 x + a_0$

Multiple terms with non-negative integer exponents

Radical Function

$y = \sqrt{x}$ or $y = \sqrt[3]{x}$

Contains a square root, cube root, or other root

1. Quadratic Functions

$ y = ax^2 + bx + c $

📌 RULE 1: Vertex Formula

The vertex $(h, k)$ is the maximum or minimum point of the parabola.

Step 1: Find the x-coordinate:
\[ h = -\frac{b}{2a} \]

Step 2: Find the y-coordinate by substituting $h$ back into the equation:
\[ k = a(h)^2 + b(h) + c \]

Vertex: $(h, k)$

📌 RULE 2: Finding Roots (x-intercepts)

Set $ y = 0 $ and factor, or use the Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The roots are the x-values where the parabola crosses the x-axis.

📌 RULE 3: Y-Intercept

Set $ x = 0 $:

\[ y = a(0)^2 + b(0) + c = c \]
The y-intercept is $(0, c)$.

📌 RULE 4: Direction of Opening

\[ a > 0 \quad \rightarrow \quad \text{Opens upward (minimum point)} \]
\[ a < 0 \quad \rightarrow \quad \text{Opens downward (maximum point)} \]

💡 Strategy — Quadratic Functions

1. Identify $a$, $b$, $c$ from the equation.
2. Use $h = -\frac{b}{2a}$ to find the vertex's x-coordinate.
3. Substitute $h$ to find $k = a(h)^2 + b(h) + c$.
4. Factor or use the quadratic formula for roots.

📝 SOLVED EXAMPLE — Quadratic

Find the vertex and roots of $y = x^2 - 4x + 3$

Step 1: Identify $a = 1$, $b = -4$, $c = 3$

Step 2: Vertex x-coordinate: $h = -\frac{-4}{2(1)} = \frac{4}{2} = \color{var(--math)}{2}$

Step 3: Vertex y-coordinate: $k = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = \color{var(--math)}{-1}$

Step 4: Vertex: $\color{var(--math)}{(2, -1)}$

Step 5: Roots: $x^2 - 4x + 3 = 0$ → $(x - 1)(x - 3) = 0$ → $x = \color{var(--math)}{1, 3}$

✅ Vertex: $(2, -1)$, Roots: $x = 1, 3$

💡 Tip: Since $a = 1 > 0$, this is a minimum (opens up).

2. Exponential Functions

$ y = a \cdot b^x $

📌 RULE 1: Growth vs. Decay

\[ b > 1 \quad \rightarrow \quad \text{Growth (population growth, investment)} \]
\[ 0 < b < 1 \quad \rightarrow \quad \text{Decay (radioactive decay, depreciation)} \]
$a$ = initial value (when $x = 0$)

📌 RULE 2: Growth/Decay Factor

If a quantity increases by $r\%$, then:
\[ b = 1 + \frac{r}{100} \]

If a quantity decreases by $r\%$, then:
\[ b = 1 - \frac{r}{100} \]

💡 Strategy — Exponential Functions

1. Identify the initial value $a$.
2. Determine if it's growth ($b > 1$) or decay ($b < 1$).
3. Substitute $x$ to find the value at any time.

📝 SOLVED EXAMPLE — Exponential

$500 invested at 4% annual growth. Find the value after 3 years.

Step 1: Initial value: $a = 500$

Step 2: Growth rate = 4% → $b = 1 + 0.04 = \color{var(--math)}{1.04}$

Step 3: $x = 3$ (years)

Step 4: $y = 500(1.04)^3 = 500(1.124864) = \color{var(--math)}{562.43}$

✅ $562.43 after 3 years

💡 Tip: Growth factor = $1 + \text{rate}$ (as a decimal). 4% → 1.04

3. Polynomial Functions

$ y = a_n x^n + ... + a_1 x + a_0 $

📌 RULE 1: End Behavior

Even degree: Both ends go the same direction.
\[ a > 0 \rightarrow \text{up on both ends} \]
\[ a < 0 \rightarrow \text{down on both ends} \]

Odd degree: Ends go in opposite directions.
\[ a > 0 \rightarrow \text{down on left, up on right} \]
\[ a < 0 \rightarrow \text{up on left, down on right} \]

📌 RULE 2: Degree

The highest exponent is the degree of the polynomial.
Example: $y = 3x^4 - 2x^2 + 1$ → Degree = 4 (even)

💡 Strategy — Polynomial Functions

1. Find the leading term (highest degree term).
2. Check the degree and leading coefficient.
3. Determine end behavior using the rules above.

📝 SOLVED EXAMPLE — Polynomial

What is the end behavior of $y = -2x^3 + x^2 - 5$?

Step 1: Leading term: $\color{var(--math)}{-2x^3}$

Step 2: Degree = 3 (odd), leading coefficient = $-2$ (negative)

Step 3: Odd degree, negative coefficient →

As $x \to \infty$, $y \to \color{var(--math)}{-\infty}$

As $x \to -\infty$, $y \to \color{var(--math)}{\infty}$

✅ As $x \to \infty$, $y \to -\infty$; as $x \to -\infty$, $y \to \infty$

💡 Tip: Odd degree, negative leading coefficient → down on right, up on left.

4. Radical Functions

$ y = \sqrt{x} $ or $ y = \sqrt[3]{x} $

📌 RULE: Domain of Square Root Functions

The expression under the square root must be $\geq 0$.

Example: $y = \sqrt{x - 5}$
\[ x - 5 \geq 0 \quad \rightarrow \quad \boxed{x \geq 5} \]

💡 Strategy — Radical Functions

1. Set the radicand (inside the root) $\geq 0$.
2. Solve for $x$ to find the domain.

📝 SOLVED EXAMPLE — Radical

Find the domain of $y = \sqrt{x + 3}$

Step 1: The radicand must be $\geq 0$: $x + 3 \geq 0$

Step 2: $\color{var(--math)}{x \geq -3}$

✅ Domain: $x \geq -3$

💡 Tip: For square roots, the expression under the root must be non-negative.

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

Find the vertex of $y = x^2 - 6x + 8$

A) $(3, -1)$

B) $(-3, -1)$

C) $(3, 1)$

D) $(6, 8)$

✓ Answer: A

$a = 1$, $b = -6$. $h = -\frac{-6}{2(1)} = 3$. $k = 9 - 18 + 8 = -1$. Vertex: $(3, -1)$

📝 Solution: $h = 3$, $k = -1$

Question 2

Find the roots of $y = x^2 - 7x + 12$

A) $x = 2, 6$

B) $x = 3, 4$

C) $x = 3, 8$

D) $x = 2, 5$

✓ Answer: B

$x^2 - 7x + 12 = 0$ → $(x - 3)(x - 4) = 0$ → $x = 3, 4$

📝 Solution: $(x - 3)(x - 4) = 0$ → $x = 3, 4$

Question 3

$500 invested at 4% annual growth. What is the value after 2 years?

A) $520

B) $530

C) $550

D) $540.80

✓ Answer: D

$y = 500(1.04)^2 = 500(1.0816) = 540.80$

📝 Solution: $500(1.04)^2 = 540.80$

Question 4

Find the domain of $y = \sqrt{x + 3}$

A) $x \geq 0$

B) $x \geq 3$

C) $x \geq -3$

D) $x > -3$

✓ Answer: C

$x + 3 \geq 0$ → $x \geq -3$

📝 Solution: $x + 3 \geq 0$ → $x \geq -3$

Question 5

What is the end behavior of $y = 3x^4 - 2x^2 + 1$?

A) As $x \to \pm\infty$, $y \to \infty$

B) As $x \to \pm\infty$, $y \to -\infty$

C) As $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$

D) As $x \to \infty$, $y \to -\infty$; as $x \to -\infty$, $y \to \infty$

✓ Answer: A

Leading term: $3x^4$ (even degree, positive coefficient) → both ends go up.

📝 Solution: Even degree, positive → both ends up.

Question 6

What is the y-intercept of $y = 2x^2 - 5x + 3$?

A) $(0, 0)$

B) $(0, 2)$

C) $(0, 3)$

D) $(0, 5)$

✓ Answer: C

Set $x = 0$: $y = 2(0)^2 - 5(0) + 3 = 3$

📝 Solution: $y = 3$ → $(0, 3)$

Question 7

A population of 200 bacteria doubles every hour. How many after 3 hours?

A) 400

B) 600

C) 800

D) 1600

✓ Answer: D

$y = 200(2)^3 = 200(8) = 1600$

📝 Solution: $200(2^3) = 200(8) = 1600$

Question 8

Which is NOT a nonlinear function?

A) $y = x^2 + 3x - 5$

B) $y = 2x - 7$

C) $y = 2^x$

D) $y = \sqrt{x}$

✓ Answer: B

$y = 2x - 7$ is a linear function. All others are nonlinear.

📝 Solution: $y = 2x - 7$ is linear (straight line).

Question 9

Find the vertex of $y = -x^2 + 4x + 1$

A) $(-2, 5)$

B) $(2, 5)$

C) $(2, 3)$

D) $(-2, 3)$

✓ Answer: B

$a = -1$, $b = 4$ → $h = -\frac{4}{2(-1)} = 2$. $k = -4 + 8 + 1 = 5$. Vertex: $(2, 5)$

📝 Solution: $a=-1$, $b=4$ → $h=2$ → $k=5$

Question 10

Find the roots of $y = x^2 - 9$

A) $x = \pm 3$

B) $x = \pm 9$

C) $x = 0, 9$

D) $x = 3$ only

✓ Answer: A

$x^2 - 9 = 0$ → $x^2 = 9$ → $x = \pm 3$

📝 Solution: $x^2 = 9$ → $x = 3$, $x = -3$

🎉 WELL DONE!

You've completed the Nonlinear Functions lesson. You now know the rules for quadratic, exponential, polynomial, and radical functions.

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