Nonlinear Functions: Quadratic and Exponential
"Nonlinear functions" is one of the four officially named question types in College Board's Advanced Math domain, the domain tied with Algebra for the largest share of the Digital SAT Math section. This free, complete lesson covers the three forms of a quadratic function and what each one reveals, finding a parabola's vertex and intercepts directly from standard form, using the discriminant to count how many real solutions an equation has, exponential growth and decay, and interpreting or comparing nonlinear function behavior from graphs and tables. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. Quadratic Function Forms
A quadratic function can be written in three equivalent forms, and each one puts a different piece of information front and center.
| Form | Equation | What it reveals directly |
|---|---|---|
| Standard form | f(x) = ax² + bx + c | The y-intercept, c |
| Factored form | f(x) = a(x − p)(x − q) | The roots (zeros, x-intercepts), p and q |
| Vertex form | f(x) = a(x − h)² + k | The vertex, (h, k) |
In every form, the sign of a tells you which way the parabola opens: if a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, the parabola opens downward and the vertex is a maximum.
a > 0 – opens upward, vertex is a minimum
a < 0 – opens downward, vertex is a maximum
Worked Example: For f(x) = 2(x − 3)² + 5, find the vertex and state whether the graph opens upward or downward.
Comparing to f(x) = a(x − h)² + k, h = 3 and k = 5, so the vertex is (3, 5). Since a = 2 is positive, the graph opens upward.
1. What is the vertex of f(x) = −3(x + 2)² + 7?
Explanation: Rewrite x + 2 as x − (−2) to match the vertex form pattern, giving h = −2 and k = 7, so the vertex is (−2, 7).
2. Does the graph of f(x) = −3(x + 2)² + 7 open upward or downward?
Explanation: Since a = −3 is negative, the parabola opens downward.
3. What are the roots (zeros) of f(x) = (x − 4)(x + 7)?
Explanation: Setting each factor equal to zero: x − 4 = 0 gives x = 4, and x + 7 = 0 gives x = −7.
4. Which form of a quadratic function most directly reveals its roots?
Explanation: In factored form, a(x − p)(x − q), the roots p and q can be read off immediately by setting each factor to zero.
5. Which form of a quadratic function most directly reveals its vertex?
Explanation: Vertex form, a(x − h)² + k, gives the vertex (h, k) directly, without any further calculation.
2. Vertex, Intercepts, and the Discriminant
You don't need to convert to vertex form to find the vertex, standard form gives you a direct formula. For f(x) = ax² + bx + c, the x-coordinate of the vertex is:
x = −b2a
Substitute this value back into the function to find the y-coordinate. The y-intercept of any quadratic in standard form is simply c, the constant term.
To find how many real solutions ax² + bx + c = 0 has, without solving it, use the discriminant, b² − 4ac:
| Discriminant value | Number of real solutions |
|---|---|
| b² − 4ac > 0 | Two real solutions |
| b² − 4ac = 0 | One real solution (a repeated root) |
| b² − 4ac < 0 | No real solutions |
Worked Example: Find the vertex of f(x) = 2x² − 8x + 3.
x = −−82(2) = 84 = 2
f(2) = 2(2)² − 8(2) + 3 = 8 − 16 + 3 = −5
The vertex is (2, −5).
1. What is the x-coordinate of the vertex of f(x) = 3x² − 12x + 5?
Explanation: x = −−122(3) = 126 = 2.
2. What is the y-intercept of f(x) = −2x² + 5x − 9?
Explanation: The y-intercept of a quadratic in standard form is the constant term, c = −9.
3. How many real solutions does 2x² − 4x + 5 = 0 have?
Explanation: Discriminant = (−4)² − 4(2)(5) = 16 − 40 = −24. Since this is negative, there are 0 real solutions.
4. How many real solutions does x² − 6x + 9 = 0 have?
Explanation: Discriminant = (−6)² − 4(1)(9) = 36 − 36 = 0. A discriminant of exactly 0 means there is 1 real solution, a repeated root.
5. How many real solutions does 3x² + 2x − 8 = 0 have?
Explanation: Discriminant = (2)² − 4(3)(−8) = 4 + 96 = 100. Since this is positive, there are 2 real solutions.
3. Exponential Growth and Decay
An exponential function has the form f(x) = a · bx, where a is the initial value (the value when x = 0) and b is the growth or decay factor. If b > 1, the function models growth; if 0 < b < 1, it models decay. A growth or decay rate expressed as a percent, r, corresponds to a factor of b = 1 + r for growth or b = 1 − r for decay.
Worked Example: A population of 200 bacteria triples every hour. Write a function for the population P after t hours, and find the population after 4 hours.
P(t) = 200(3)t
P(4) = 200(3)⁴ = 200(81) = 16,200
1. A car valued at $24,000 depreciates by 15% each year. Which function models its value V after t years?
Explanation: A 15% decrease each year corresponds to a decay factor of 1 − 0.15 = 0.85, giving V(t) = 24,000(0.85)t.
2. A population of 500 bacteria doubles every 3 hours. Which function models the population P after t hours?
Explanation: The population doubles (factor of 2) every 3 hours, so the exponent must equal 1 whole "doubling period" when t = 3, which happens when the exponent is t/3.
3. In the exponential function y = 800(1.06)x, what does 1.06 represent?
Explanation: Writing 1.06 as 1 + 0.06 shows it's a growth factor corresponding to a 6% increase each period.
4. An investment of $2,000 grows at 4% annual interest, compounded yearly. What is its value after 3 years?
Explanation: V = 2,000(1.04)³ = 2,000(1.124864) = $2,249.73.
5. Which of the following represents exponential decay?
Explanation: A base between 0 and 1 represents decay. Only 0.7 fits that range; 1.2 and 2 are both greater than 1 (growth), and the last option isn't exponential at all.
4. Interpreting and Comparing Nonlinear Functions
On the Digital SAT, nonlinear function questions often show a table, a graph, or a real-world description rather than asking you to graph something yourself. Recognizing the pattern, and reading off key values, is the core skill being tested.
Exponential growth – rises slowly, then steeply
Exponential decay – falls quickly, then levels off toward 0
Worked Example: A ball is thrown upward, and its height h, in feet, after t seconds is modeled by h(t) = −16t² + 64t. What is the maximum height reached, and when does it occur?
The maximum occurs at the vertex. The t-coordinate is t = −642(−16) = 6432 = 2 seconds.
h(2) = −16(2)² + 64(2) = −64 + 128 = 64 feet
1. A table shows y-values doubling every time x increases by 1. What type of function does this best represent?
Explanation: A constant multiplying factor (doubling) for each equal step in x is the signature of exponential growth, not a constant additive difference, which would indicate a linear function.
2. As x increases without bound, what happens to y in the exponential decay function y = 100(0.5)x?
Explanation: Since 0.5 is a decay factor, repeated multiplication by 0.5 shrinks the value closer and closer to 0, but never reaches or goes below it.
3. A ball's height h, in feet, after t seconds is modeled by h(t) = −16t² + 64t. What is the maximum height reached?
Explanation: This is the worked example above: the vertex occurs at t = 2 seconds, giving a maximum height of h(2) = 64 feet.
4. Using the height function from the previous problem, after how many seconds does the ball reach its maximum height?
Explanation: The vertex t-coordinate is t = −642(−16) = 2 seconds.
5. A quadratic function has a negative leading coefficient. As x approaches positive or negative infinity, what happens to y?
Explanation: A negative leading coefficient means the parabola opens downward, so both arms of the graph fall toward negative infinity as x moves away from the vertex in either direction.
Common Mistakes to Avoid
- Reading the vertex coordinates with the wrong sign for h. In a(x − h)² + k, a plus sign inside the parentheses, like (x + 2)², means h = −2, not 2. Rewrite addition as subtracting a negative before identifying h.
- Forgetting the negative sign in the vertex formula. The formula is x = −b/(2a), not b/(2a). Dropping that negative sign is one of the most common errors when finding a vertex from standard form.
- Mixing up growth and decay factors. A growth rate of r% gives a factor of 1 + r (as a decimal); a decay rate of r% gives a factor of 1 − r. Writing the percent itself as the base, instead of 1 plus or minus the percent, is a frequent error.
- Misreading the discriminant's sign. A positive discriminant means two real solutions, zero means one, and negative means none, in that order. It's easy to reverse the positive and negative cases under time pressure.
- Assuming every nonlinear pattern in a table is quadratic. A constant ratio between consecutive y-values signals exponential behavior; a constant second difference signals quadratic behavior. Check which pattern actually holds before choosing a model.
Frequently Asked Questions
Do I need to memorize all three quadratic forms and how to convert between them?
You should recognize what each form reveals and be comfortable reading off information directly from whichever form a question gives you. Converting between forms, like completing the square to go from standard to vertex form, is useful but tested less often than simply extracting information from the form already provided.
How is exponential growth different from quadratic growth?
A quadratic function grows by a changing additive amount, the differences between consecutive outputs increase by a constant amount. An exponential function grows by a constant multiplicative factor, each output is a fixed multiple of the previous one. Over the long run, exponential growth always eventually outpaces quadratic growth.
Why does the discriminant tell you the number of solutions without solving the equation?
The discriminant is the expression under the square root in the quadratic formula. A negative number has no real square root, giving no real solutions; zero has exactly one square root value (itself), giving one repeated solution; and a positive number has two square roots (positive and negative), giving two solutions.
Where can I practice more problems like these?
The Nonlinear Functions quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.