Pre-calculus 12 tutor for BC students, live online

Source: BC Pre-calculus 12 curriculum, content and curricular competencies, in a copy fetched raw on 23 October 2026; the Calculus 12 reference is from the BC Calculus 12 page, fetched 22 October 2026. Modern Age Coders has no connection with the BC ministry of education or any school district.

The problem. Let f(x) = (2x + 7) / (x + 3). (a) Write f as a transformation of y = 1/x. (b) State its asymptotes, domain, range and intercepts. (c) Find the inverse function and check it.

1. Divide to expose the parent. 2x + 7 = 2(x + 3) + 1, so f(x) = 2 + 1/(x + 3). That is y = 1/x moved 3 units left and 2 units up. No stretch and no reflection: the coefficient of the fraction is 1.
2. Read the asymptotes from the moves. The parent has asymptotes x = 0 and y = 0, so f has x = −3 and y = 2. Domain: every real number except −3. Range: every real number except 2.
3. Find the intercepts. Setting x = 0 gives f(0) = 7/3. Setting f(x) = 0 means 2x + 7 = 0, so x = −3.5.
4. Swap and solve for the inverse. Write x = 2 + 1/(y + 3). Then x − 2 = 1/(y + 3), so y + 3 = 1/(x − 2), and f⁻¹(x) = 1/(x − 2) − 3.
5. Check with the graph’s logic. An inverse reflects the graph in y = x, so its asymptotes swap: x = 2 and y = −3. They do. Its domain is the original range, all reals except 2.
6. Check with numbers. Feed values through f and then through the inverse; each should come back unchanged.

# Check that two functions undo each other, on a table of values (ours)
def f(x):
    return (2*x + 7) / (x + 3)

def f_inv(x):
    return 1 / (x - 2) - 3

for x in [-5, -2, 1, 1.5, 10]:
    y = f(x)
    print(f"x = {x:>5}   f(x) = {y:8.4f}   f_inv(f(x)) = {f_inv(y):8.4f}")

x =    -5   f(x) =   1.5000   f_inv(f(x)) =  -5.0000
x =    -2   f(x) =   3.0000   f_inv(f(x)) =  -2.0000
x =     1   f(x) =   2.2500   f_inv(f(x)) =   1.0000
x =   1.5   f(x) =   2.2222   f_inv(f(x)) =   1.5000
x =    10   f(x) =   2.0769   f_inv(f(x)) =  10.0000

1. Transformations. The point (4, 5) is on the graph of y = f(x). Where does it land on y = −3f(2(x − 1)) + 4?
2. Inverse. Find the inverse of f(x) = (x − 2) / (x + 5).
3. Same base. Solve 4^{x + 1} = 8^{2x − 1}.
4. Different bases. Solve 3^x = 5^{x − 2}, exactly and to three decimal places.
5. Logarithm laws. Solve log₃ (x + 6) + log₃ x = 3, and test each candidate in the original equation.
6. Infinite series. A geometric series has first term 12 and common ratio 3/4. Write its general term and find the sum to infinity.
7. Sigma notation. Evaluate the sum of 5 · 2^{k − 1} for k = 1 to 6.
8. Factor and remainder theorems. Factor P(x) = x³ − 4x² + x + 6 fully, then find the remainder when P(x) is divided by x − 4.
9. Rational function. For y = (x² − 4) / (x² − x − 2), find every asymptote and every point discontinuity.
10. Trig graph. Describe y = −2 cos(3(x + π/4)) − 1: its amplitude, its period, its horizontal shift and its range.
11. Trig equation. Solve 2 cos² x − cos x − 1 = 0 for 0 ≤ x < 2π.
12. Identity. Show that (1 − cos 2x) / sin 2x = tan x, and state any values of x where the statement is not defined.

1. Ask for the assessment plan. Unit tests, quizzes, a final exam, calculator rules, and how each piece is weighted.
2. Make a parent-function sheet. One small graph each for absolute value, radical, reciprocal, exponential, logarithmic, sine and cosine, with key points and asymptotes marked.
3. Practise transformations on points, not whole graphs. Take three key points, apply the change, plot them. It is faster and catches order-of-operations errors.
4. Pair every exponential with a logarithm. For each exponential equation solved, solve its logarithmic twin. The two units then reinforce each other.
5. Check every candidate root. After any logarithm, radical or squaring step, substitute each answer into the original equation. Write the check down.
6. Build an identity list from memory weekly. Pythagorean, quotient, reciprocal, double angle, sum and difference: write them out without notes, then use two in a fresh problem.
7. Solve trig equations twice. Once on the interval asked, once over all real numbers, so the general solution is never a surprise.
8. Keep a mistakes page by unit. Note whether each error was a slip or a misunderstanding; misunderstandings get re-taught first.
9. Do a mixed set every week from week three. A final mixes units; practice should too, like our twelve questions above.
10. Seven days out, do one full paper against the clock. After that, revise only from the mistakes page; no new topics.