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Nonlinear functions: SAT Math notes
Working with functions whose graphs curve, mostly exponential models and quadratics: writing them from a description, reading key points from their graphs, and explaining what their numbers mean.
Study it with worked examples and 8 practice questionsPractise Nonlinear functions questions
What this topic tests
A nonlinear function does not change by the same amount at each step, so its graph curves. On the SAT the two big families are exponential functions, which multiply by the same factor at each step, and quadratics, whose graphs are parabolas. You will also see a few cubics and other polynomials.
The questions come in a few flavours: write the right equation from a story, explain what a number in an equation means, find a key point such as the vertex or an intercept, or plug a value into a function. Many are set in real situations like population growth, a ball in the air, or a company's revenue, so always ask yourself what x and y stand for.
Desmos is especially useful in this skill. Type a function and click its graph to see the vertex, the intercepts and other points, or define f(x) and type f(3) to evaluate it. Still learn what each part of an equation means, because many questions ask you to interpret rather than calculate.
Key ideas
- Linear functions add the same amount at each step. Exponential functions multiply by the same factor at each step.
- Exponential model: f(x)=a·bx. a is the starting value and b is the growth factor. If b>1 it grows; if 0<b<1 it decays.
- Percent to factor: up p% gives b=1+p/100, and down p% gives b=1-p/100.
- If the change happens once every k units of time, the exponent is x/k.
- Vertex form a(x-h)2+k has its vertex at (h,k). The vertex sits halfway between the two x-intercepts.
- In ax2+bx+c, the vertex is at x=-b/2a and the y-intercept is c.
- The y-intercept of any function is f(0). An x-intercept is a value of x where f(x)=0.
The 4 kinds of Nonlinear functions question
Sorted from every Nonlinear functions question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Exponential growth and decay
How to spot it. A quantity changes by the same percent or the same factor over equal time steps: 'doubles every 25 minutes', 'decreases by 20% each year', 'increases by 70% every 2 days'. You pick the right equation, explain a number in f(x)=a·bx, find a percent change, or switch the time unit.
How to answer it:
- Use the template f(x)=a·bx/k: a is the amount at the start (when x=0), b is what you multiply by at each step, and k is how long one step takes.
- Turn the percent into b. A p% increase means b=1+p/100, and a p% decrease means b=1-p/100. So a 20% decrease gives 0.80, 150% more gives 2.5, and doubling gives 2.
- Match the exponent to the time step. If the change happens every 2 days and x counts days, the exponent is x/2. If a model is in years and you need months, replace t with m/12.
- To interpret, remember that a is the starting value (the y-intercept) and b tells you the percent change per step. For example, b=0.90 means a 10% drop each step.
- Desmos: graph the model and click points or open a table to check values, for example to see when it passes a target.
The trap. Using the percent itself as the base. A 20% drop each year means multiplying by 0.80, not 0.20, and growing by 70% means multiplying by 1.70, not 0.70. Also check the starting point: a must be the value at time zero, which is not always the first number in the story.
2. Parabolas and polynomials: vertex, zeros, max and min
How to spot it. A quadratic, or a polynomial in factored form, and a question about its graph: the vertex, the maximum or minimum, the x-intercepts, the y-intercept, or what one of those means in a story about height, revenue or profit.
How to answer it:
- Read the form you are given. Vertex form a(x-h)2+k shows the vertex (h,k). Factored form a(x-r)(x-s) shows the zeros r and s. Standard form ax2+bx+c shows the y-intercept c.
- Use the sign of a: positive means the parabola opens up and has a minimum, negative means it opens down and has a maximum.
- The vertex is halfway between the zeros, at x=(r+s)/2, or at x=-b/2a in standard form. The graph is symmetric about that vertical line, so points at equal heights come in mirror pairs.
- For the zeros of any polynomial, set each factor equal to zero. If p(a)=0, then x-a is a factor.
- In a story, the x of the vertex is when or where the best (or worst) happens, the y of the vertex is that best value, the y-intercept is the starting value, and a positive x-intercept is when the quantity reaches zero, like a ball hitting the ground.
- Desmos: type the function and click the graph. Desmos marks the vertex and the intercepts as gray points with their coordinates.
The trap. Mixing up the input and the output of the vertex. 'At what angle is airflow greatest?' wants h, the x-coordinate, while 'What is the greatest airflow?' wants k. Also, the factor x-14 gives a zero at positive 14, not -14.
3. Evaluating functions and reading graphs
How to spot it. You are given a function rule, a table or a graph and asked for an output like f(-1), the input that gives a certain output, the y-intercept, or what a statement like f(5)=648,650 means. Sometimes the rule has an unknown constant that you must find first.
How to answer it:
- To find f(a), replace every x with a in brackets. For f(-1), write (-1)2, not -12.
- Follow the order of operations: powers, then multiplying and dividing, then adding and subtracting. Work out the top and the bottom of a fraction separately.
- If you know the output and need the input, set the rule equal to that output and solve.
- If the rule has an unknown constant, use the fact you are given (like p(c)=10) to find it first, then evaluate.
- The y-intercept is always f(0); on a graph it is where the curve crosses the vertical axis. In a story, f(a)=b means that when the input is a, the output is b.
- Desmos: define the function, for example f(x)=(x^2-6x+3)/(x-1), then type f(-1) on the next line to see the value.
The trap. Sign errors with negative inputs: (-1)2=1 and -6(-1)=+6. In interpretation questions, don't swap the input and the output. In h(1.4)=1.64, 1.4 is the time and 1.64 is the height.
4. Building a nonlinear model from a description
How to spot it. A word problem about area, volume, or the product of two numbers, where one measurement is described using another, such as 'the length is 5 feet longer than the width'. You build a quadratic (or cubic) and then either pick it from the choices or solve it.
How to answer it:
- Name the unknown with one letter, usually the simpler quantity, and write every other quantity using it. For example, width w and length w+5.
- Write the formula: the area of a rectangle is length times width, the area of a triangle is 1/2bh, the volume of a box is length times width times height, and 'product' means multiply.
- If the question wants the function, simplify and match it to a choice. If it gives a value, set the formula equal to that value and rearrange to ax2+bx+c=0.
- Solve by factoring, by the quadratic formula, or by testing the choices. Throw out negative lengths. In Desmos, you can graph y = your expression and y = the given value and click the crossing point with a positive x.
- Answer the question that was asked. It may want the length even though you solved for the width.
The trap. Answering for the wrong quantity. If you let w be the width but the question asks for the length, add the extra amount at the end, because the width is often one of the wrong choices. Also remember the 1/2 in the triangle area formula.
Practise Nonlinear functions
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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