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Nonlinear functions: quadratics, exponentials, others
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Read the last line first
- Start:
- Start with the question sentence at the end: does it ask for a value, a coordinate, a formula, or what a number means?
- Stop:
- Stop when you know exactly what is asked (for example 'maximum height').
- Skip:
- Skip the long story sentences until you know what you need from them.
- Why:
- In question 5 the starting height 80 is one of the answer options, but the question wants the highest point, not the start. Knowing the target stops that trap.
Used in questions 3, 5
2. Vertex x = -b ÷ (2a), then plug back
- Start:
- Find a and b in the standard form.
- Stop:
- Stop once you have the y-value at the vertex.
- Skip:
- Skip drawing the graph and skip expanding anything.
- Why:
- Questions 3 and 5 are both solved by the same two steps. In Desmos you can also type the function and click the lowest or highest point to check.
Used in questions 3, 5
3. Two zeros means the vertex is in the middle
- Start:
- Write the zeros from the factored form.
- Stop:
- Stop after finding a using any extra point.
- Skip:
- Skip multiplying out the brackets.
- Why:
- In question 7 one extra point gives a, and the factored form gives the rest. Expanding wastes time and adds mistakes.
Used in question 7
4. Discriminant for 'no real solutions' or 'exactly one'
- Start:
- Write b² - 4ac.
- Stop:
- Stop when you have the inequality (negative for none, zero for one, positive for two).
- Skip:
- Skip solving the quadratic.
- Why:
- Question 8 only asks about how many solutions. Test the boundary value 16 first: it is always the trap option.
Used in question 8
5. Exponential: divide to find the multiplier
- Start:
- Divide the later value by the earlier value.
- Stop:
- Stop when you know b (take the root if there are several steps).
- Skip:
- Skip trying to solve for a and b at the same time with big algebra.
- Why:
- In question 6, 108 ÷ 12 = 9 gives b² at once. Then a is easy.
Used in question 6
6. Count the time periods in a story
- Start:
- Find the 'every ___ hours' phrase.
- Stop:
- Stop when the exponent is time ÷ period.
- Skip:
- Skip any option with the wrong direction (growth vs decay).
- Why:
- Question 4 can be solved by testing t = 3 in each option. Doubling at t = 3 must give 400.
Used in questions 2, 4
7. Plug in a simple number to test options
- Start:
- Choose a number that is easy, such as x = 0 or x = 1.
- Stop:
- Stop when only one option matches.
- Skip:
- Skip fully simplifying each option.
- Why:
- Questions 1 and 2 are direct plug-in questions, and in function-matching questions one or two points remove most options. Put the number in Desmos to check fast.
Used in questions 1, 2, 4
8. Equal outputs in a table show the vertex
- Start:
- Look for two x-values in the table that give the same output.
- Stop:
- Stop when you find the middle of those two x-values: that is the axis of symmetry.
- Skip:
- Skip the guessing of a, b and c from scratch, and skip the biggest or smallest table value.
- Why:
- In questions 14 and 15 the table never shows the real maximum or minimum. The table maximum 400 and the table minimum -5 are traps. The vertex is halfway between equal outputs.
Used in questions 14, 15
9. Constant ratio means exponential, constant difference means line
- Start:
- Divide each table value by the one before it, then subtract them.
- Stop:
- Stop when the ratios are all equal (exponential) or the differences are all equal (linear).
- Skip:
- Skip fitting a graph to the numbers.
- Why:
- Question 9 has a ratio of 1.5 each week, so the answer is exponential. In question 16, treating the graph as a line gives the trap 29.
Used in questions 9, 16
10. Graph questions: use the labeled points
- Start:
- Find every point or line that is labeled on the graph (dots, intercepts, a dashed asymptote).
- Stop:
- Stop when each label has given you one equation or one fact.
- Skip:
- Skip reading unlabeled values by eye, and skip the legend decoration.
- Why:
- Questions 10, 12 and 16 all give labeled points. In question 12 the vertex plus one more point gives a. In question 16 the dashed line gives c at once.
Used in questions 10, 12, 16