Year 13 in England
12th grade algebra
18 12th grade algebra skills, each with a worked example, free practice that checks every answer, and a printable worksheet. Start with a mixed review, or jump to the skill you need.
Mixed review
Ten questions drawn from every skill below.
Exponential and logarithmic equations
CCSS HSF.LE.A.4CCSS HSF.BF.B.5
Example
Solve: 2^(x − 1) = 2
- 2 = 2¹
- x − 1 = 1
- x = 2
Answer: x = 2
Builds on: Logarithms
Continuous growth and the number e
CCSS HSF.LE.A.4CCSS HSF.LE.B.5
Example
4500 is invested at 4% interest compounded continuously, A = Pe^(rt). Find A after 13 years, to 2 decimal places.
- rt = 0.52
- A = 4500e^0.52 ≈ 7569.12
Answer: 7569.12
Builds on: Exponential models, Logarithms
Polynomial functions: zeros and end behaviour
CCSS HSA.APR.B.3CCSS HSF.IF.C.7.C
Example
f(x) = −(x − 2)(x − 3)(x + 3). Find all the zeros of f, separated by commas.
- x = −3, 2, 3
Answer: x = -3 or x = 2 or x = 3
Builds on: Remainder and factor theorems
Rational functions and asymptotes
CCSS HSF.IF.C.7.DCCSS HSA.APR.D.6
Example
Find the vertical asymptote of f(x) = (3x + 2)/(3x + 6). Give the x-value.
- 3x + 6 = 0
- x = −2
Answer: -2
Absolute value equations and inequalities
CCSS HSA.REI.B.3CCSS HSA.CED.A.1
Example
Solve: |2x + 20| = 8. Give both solutions, separated by a comma.
- 2x + 20 = 8 → x = −6
- 2x + 20 = −8 → x = −14
Answer: x = -14 or x = -6
Sigma notation and infinite geometric series
CCSS HSA.SSE.B.4
Example
Find the sum to infinity of the geometric series with first term 5 and common ratio 2/5.
- S∞ = 5 ÷ 3/5
- = 25/3
Answer: 25/3
Matrices: operations and determinants
CCSS HSN.VM.C.8CCSS HSN.VM.C.10CCSS HSN.VM.C.12
Example
Find the determinant of the matrix [[8, −1], [−2, 3]].
- 8 × 3 − (−1) × (−2)
- = 24 − 2 = 22
Answer: 22
Systems of three equations
CCSS HSA.REI.C.6
Example
Solve: x + y + z = 6, x − y + 2z = −4, 2x + y − z = 14. What is z?
- 2x + 3z = 2
- 3x + z = 10
- x = 4, y = 4, z = −2
Answer: -2
Inverse functions (non-linear)
CCSS HSF.BF.B.4.ACCSS HSF.BF.B.5
Example
f(x) = 3/(x − 5). Find f⁻¹(x).
- x = 3/(y − 5) → y − 5 = 3/x
- f⁻¹(x) = 3/x + 5
Answer: 3/x+5
Builds on: Composite and inverse functions, Logarithms
Leads to: Relate a function's graph to its inverse
Solve equations with e and natural logarithms
CCSS HSF.LE.A.4England: A-level Pure Mathematics
Example
Solve eˣ = 16. Give x to 3 decimal places.
- x = ln 16
- x ≈ 2.773
Answer: 2.773
Evaluate logarithms with the change-of-base formula
CCSS HSF.LE.A.4England: A-level Pure Mathematics
Example
Use the change-of-base formula to evaluate log₇ 114 to 3 decimal places.
- log 114 ÷ log 7 ≈ 2.434
Answer: 2.434
Builds on: Evaluate logarithms without a calculator
Find vertical and horizontal asymptotes
CCSS HSF.IF.C.7.DEngland: A-level Pure Mathematics
Example
f(x) = (4x − 9)/(x − 1). Find the vertical asymptote x = ?
- x = 1
Answer: 1
Evaluate sums written in sigma notation
CCSS HSA.SSE.B.4England: A-level Pure Mathematics
Example
Evaluate Σ (from r = 1 to 11) of (−2r + 2).
- first = 0, last = −20
- 11(0 + (−20))/2 = −110
Answer: -110
Find the sum to infinity of a geometric series
CCSS HSA.SSE.B.4England: A-level Pure Mathematics
Example
Find the sum to infinity of the geometric series with first term 36 and common ratio 3/4. Give a fraction if needed.
- 1 − (3/4) = 1/4
- S∞ = 36 ÷ 1/4 = 144
Answer: 144
Builds on: Sum a finite geometric series
Find a coefficient in a binomial expansion
CCSS HSA.APR.C.5England: A-level Pure Mathematics
Example
Find the coefficient of x in the expansion of (1 + x)⁴.
- 4C1 = 4
Answer: 4
Builds on: Permutations and combinations, Expand three binomials
Decide whether a function is even, odd or neither
CCSS HSF.BF.B.3England: A-level Pure Mathematics
Example
Is f(x) = x³ − x² + 3 even, odd or neither?
- Neither
Answer: Neither
Builds on: Transformations of functions
Relate a function's graph to its inverse
CCSS HSF.BF.B.4.CEngland: A-level Pure Mathematics
Example
The point (−1, −4) is on the graph of y = f(x). Which point must be on the graph of y = f⁻¹(x)? Type it like (2, −1).
- (−4, −1)
Answer: (−4, −1)
Builds on: Inverse functions (non-linear)
Compare compounding periods and continuous interest
CCSS HSF.LE.A.2CCSS HSA.SSE.B.3.CEngland: A-level Pure Mathematics
Example
$5000 is invested at 5% a year, compounded monthly. Use A = P(1 + r/n)^(nt) to find the value after 3 years, to 2 decimal places.
- 5000(1 + 0.05/12)^36
- ≈ 5807.36
Answer: 5807.36
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