Introduction to Nonlinear Functions
Welcome to the world of Nonlinear Functions! Up until now, you might have spent a lot of time with straight lines (linear functions). But life isn't always a straight line—sometimes it curves, jumps, or speeds up rapidly. In this chapter, we will explore functions where the graph is anything but a straight line. These are essential for the SAT because they help us model everything from the path of a kicked soccer ball to the way money grows in a bank account. Don't worry if these curves look intimidating at first; we’re going to break them down into simple, manageable shapes.
1. Quadratic Functions: The "U" Shape
The most famous nonlinear function on the SAT is the Quadratic Function. Its graph is called a parabola, which looks like a "U" or an upside-down "U."
The Three Forms You Must Know
The SAT will test you on three different ways to write a quadratic equation. Each one "hides" a different piece of information in plain sight:
1. Standard Form: \(f(x) = ax^2 + bx + c\)
- The value c is the y-intercept (where the graph hits the vertical axis).
- If a is positive, the "U" opens up (like a smile). If a is negative, it opens down (like a frown).
2. Vertex Form: \(f(x) = a(x - h)^2 + k\)
- This is the most helpful form! The vertex (the very tip or bottom of the curve) is at the point (h, k).
- Quick Tip: Notice the minus sign in front of the h. If the equation is \((x - 3)^2\), the h is actually positive 3!
3. Factored Form: \(f(x) = a(x - r_1)(x - r_2)\)
- The numbers \(r_1\) and \(r_2\) are the zeros or roots. These are the spots where the graph crosses the x-axis.
Analogy: Think of a quadratic function like throwing a ball in the air. It starts at a certain height (y-intercept), reaches a peak (vertex), and eventually hits the ground (roots).
Common Mistake to Avoid: When moving from factored form to the vertex, remember that the vertex is always exactly halfway between the two roots! If your roots are at 2 and 6, the vertex must be at \(x = 4\).
Key Takeaway: Quadratic functions involve an \(x^2\) term. Look at the form of the equation to quickly find the vertex, y-intercept, or roots without doing extra math.
2. Exponential Functions: The "Fast Growers"
If a quadratic function is a curve, an Exponential Function is a rocket ship. These functions represent things that double, triple, or shrink by a percentage over time.
The Basic Formula
\(f(x) = a(b)^x\)
a: This is the initial value (what you start with when \(x = 0\)).
b: This is the growth factor.
Did you know?
- If b is greater than 1, the graph is "growing" (Exponential Growth).
- If b is between 0 and 1 (like 0.5), the graph is "shrinking" (Exponential Decay).
Real-World Example: If you have \$100 and it doubles every year, your equation is \(f(x) = 100(2)^x\). After 1 year, you have \$200. After 2 years, you have \$400. It adds up fast!
Quick Review: Linear functions add the same amount every time (1, 2, 3, 4...). Exponential functions multiply by the same amount every time (2, 4, 8, 16...).
Key Takeaway: In exponential functions, the variable \(x\) is in the exponent. These graphs never cross the horizontal line they are "hugging" (called an asymptote).
3. Polynomial and Absolute Value Functions
The SAT also includes other shapes that follow specific rules.
Absolute Value Functions
The graph of \(f(x) = |x|\) always looks like a sharp "V".
- The "tip" of the V works just like the vertex of a quadratic.
- Memory Aid: Absolute Value looks like a V.
Polynomials (Higher Degrees)
Equations like \(f(x) = x^3 - 2x\) are called polynomials.
- The "degree" is the highest exponent.
- The degree tells you the maximum number of times the graph can cross the x-axis. A degree of 3 (cubic) can cross up to 3 times.
Key Takeaway: For higher-degree polynomials, focus on the zeros (where \(f(x) = 0\)). If an equation is \((x-2)(x+3)(x-5)\), the graph crosses the x-axis at 2, -3, and 5.
4. Radical and Rational Functions
These functions have special "restricted" zones where the graph simply cannot go.
Radical Functions
These involve square roots, like \(f(x) = \sqrt{x}\).
- You cannot take the square root of a negative number in the "real" world of SAT graphing.
- This means the graph has a "starting point" and only goes in one direction.
Rational Functions
These are fractions with an \(x\) in the denominator, like \(f(x) = \frac{1}{x-3}\).
- The Golden Rule: You can never divide by zero!
- In the example above, \(x\) cannot be 3. On the graph, this creates a vertical "invisible wall" called an asymptote that the graph will never touch.
Key Takeaway: When you see a fraction, look at the bottom. Whatever makes the bottom zero is where the graph breaks!
5. Systems of Nonlinear Equations
Sometimes the SAT will give you two equations—maybe a circle and a line, or a parabola and a line—and ask where they meet.
How to solve them:
1. Substitution: If one equation is \(y = x^2\) and the other is \(y = 2x + 3\), set them equal to each other: \(x^2 = 2x + 3\).
2. Solve for x: Move everything to one side (\(x^2 - 2x - 3 = 0\)) and factor.
3. Number of Solutions:
- If the line crosses the curve twice, there are 2 solutions.
- If the line just grazes the edge of the curve, there is 1 solution.
- If they never touch, there are 0 solutions.
Common Mistake: When you solve for \(x\), don't forget to plug it back into one of the original equations to find the \(y\) value! A "solution" is a coordinate pair \((x, y)\).
Key Takeaway: Solutions to a system are just the points where the two graphs intersect (cross each other).
Final Quick Tips for Success
- Check the scale: SAT graphs often have different scales on the x and y axes. Always look at the numbers before picking an answer.
- Plug and Chug: If you are stuck on a nonlinear question, pick a point \((x, y)\) from the graph and plug it into the answer choices. If the math doesn't work, that's not the right equation!
- Don't Panic: Nonlinear functions are just shapes. Once you recognize the shape (U for quadratic, V for absolute value, Curve for exponential), you're halfway to the answer.