You can say what a function is heading toward near a point, even when it never actually gets there.
Weeks 13-14
Precalculus course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Function analysis
Weeks 1-2
You can take functions apart, glue them together, and undo one with its inverse. You can also describe every shift, stretch and flip you see in a graph.
Find f(g(x)) when f(x) = x squared and g(x) = x - 3.
Find the inverse of f(x) = 2x + 7 by swapping x and y.
Say how y = -f(x + 2) moves and flips the graph of f.
Worked example
Find the inverse of f(x) = 2x + 7
Write y = 2x + 7, then swap the letters: x = 2y + 7
x - 7 = 2y
AnswerThe inverse is (x - 7)/2
Most common mistakeMistaking the inverse for the reciprocal: writing 1/(2x + 7), which sends 3 to about 0.077 instead of back to -2.
You can match a polynomial or rational formula to a real situation and read what its graph is telling you. You can also find holes, asymptotes and turning behaviour.
Find the horizontal asymptote of (3x squared + 1)/(x squared - 5) by comparing lead terms.
Spot a hole where the same factor cancels top and bottom.
Build a cubic volume model for a box made by cutting corners out of a sheet.
Worked example
Find the horizontal asymptote of y = (3x squared + 1)/(x squared - 5)
Top and bottom are both degree 2, so the degrees tie
Divide the leading coefficients: 3/1
Answery = 3
Most common mistakeSetting the numerator to zero instead: solving 3x squared + 1 = 0, finding nothing, and concluding there is no asymptote when the graph clearly levels off at y = 3.
You can find sine, cosine and tangent for any angle and sketch the waves they trace. You can also read amplitude, period and shift straight from a formula.
Convert 150 degrees into radians.
Find the period of y = 4 sin(3x).
Give the exact value of cos(5 pi / 6) without a calculator.
Worked example
Find the amplitude and period of y = 4 sin(3x)
Amplitude is the size of the 4 out front
Period = 2 pi / 3
AnswerAmplitude 4, period 2 pi / 3
Most common mistakeMultiplying by the 3 rather than dividing: reporting a period of 6 pi, nine times too long, so the sketch shows one wave where there should be nine.
You can recognise a circle, ellipse, parabola or hyperbola from its equation and find its centre, vertices and foci. You can also complete the square to tidy a messy equation up.
Sort an equation by looking at the signs and coefficients of the squared terms.
Complete the square on x squared + 6x + y squared = 7 to find the centre.
Find the foci of an ellipse using c squared = a squared - b squared.
Worked example
Find the centre and radius of x squared + 6x + y squared = 7
Complete the square: add 9 to both sides
(x + 3) squared + y squared = 16
AnswerCentre (-3, 0), radius 4
Most common mistakeAdding the 9 to the left side only: writing (x + 3) squared + y squared = 7 and reporting a radius of about 2.65 instead of 4.
You can define a sequence with a direct formula or a step-by-step rule, and add geometric terms exactly. You can also tell whether an endless geometric sum settles on a number.
Turn the recursive rule a(n) = 2a(n-1) + 1 with a(1) = 3 into a list of terms.
Add the infinite series 8 + 4 + 2 + 1 and so on.
Write the sum of the first 12 terms using sigma notation.
Worked example
Add the infinite series 8 + 4 + 2 + 1 + ...
First term a = 8, ratio r = 1/2
S = a/(1 - r) = 8/0.5
Answer16
Most common mistakeUsing S = a/(1 - r) when the ratio is bigger than 1: for 2 + 4 + 8 + ... it returns -2, a negative total for a sum of positive numbers that never stops growing.
Before timing yourself, check whether you can explain Function analysis from a blank page. Then connect it to Polynomial and rational models. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Precalculus calculator can test calculations and representations used in Precalculus. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Precalculus practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Function analysis if you are following the full sequence. If that unit feels automatic, open the Precalculus practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Precalculus test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Precalculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
