Transforming Functions
How does writing a relationship as a function let us describe, and reshape, change?
The MYP criteria this unit is marked against.
- Knowing and Understanding
- Investigating Patterns
- Communicating
- Applying Mathematics in Real-life Contexts
What this unit covers
The sub-topics taught, in the order the department teaches them.
- Set notation for domain & range
- Definition of function and relation, vertical line test, function notation
- Inverse & composite functions (EXT)
- Solving quadratic equations
- Quadratic functions
- Transformations of functions
Learning objectives
What students should be able to do by the end of this unit.
Functions
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- Understand the definition of a function and a relation; use domain and range.
- Apply the vertical line test to determine whether a relation is a function.
- Use function notation: evaluate $f(a)$, solve $f(x) = k$, and interpret in context.
- Find the largest natural domain of a real-valued function involving fractions and square roots.
- Determine inverse functions both algebraically and graphically.EXT
- Use composite functions with $f \circ g$ notation; find the domain of a composition.EXT
- Recognise and sketch absolute-value and step functions.EXT
- Explore Euler's number $e$ and the function $e^x$.ENR
Solving Quadratics
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- Factorise quadratic expressions (monic and non-monic) and solve by the null factor law.
- Solve quadratic equations using the quadratic formula.
- Complete the square and convert between expanded, factorised and vertex forms.
Quadratic Graphs and Modelling
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- Identify key features of a quadratic graph: vertex, axis of symmetry, $x$- and $y$-intercepts.
- Use the GDC to explore quadratic features, including optimisation.
- Solve real-world quadratic modelling problems (projectile, area, profit).
- Solve quadratic inequalities by sketch or sign-table.EXT
- Use the discriminant $b^2 - 4ac$ to determine the nature of roots.EXT
- Apply the discriminant to tangency / intersection problems with a line.EXT
Transformations of Functions
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- Describe and perform vertical translations $y = f(x) + c$ and horizontal translations $y = f(x - h)$.
- Describe and perform vertical dilations $y = a \cdot f(x)$ and horizontal dilations $y = f(ax)$.
- Describe and perform reflections in the $x$-axis ($y = -f(x)$) and in the $y$-axis ($y = f(-x)$).
- Apply a sequence of transformations to a graph and state the new key features.
- Match transformed graphs to their algebraic form, and vice versa.
- Use vertex form $y = a(x - h)^2 + k$ to read off translations and dilations of a quadratic.
- Apply transformations to known parent functions (linear, quadratic, exponential).
Practice & review
Targeted practice for each skill, hosted on Dr Frost Maths. Tick when comfortable.
Domain and Range
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- Practice on DFM →#343Inequalities for expressions given a restricted domain on the variables
- Practice on DFM →#437Domain and range of simple functions
Functions
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- Practice on DFM →#195Function machines involving decimals, negative numbers and algebraic inputs
- Practice on DFM →#432Function notation and calculating outputs or inputs or a function
Advanced Functions
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- Practice on DFM →#433Graph representation of functions
- Practice on DFM →#435Composite functions (excluding exponentials and modulus)
- Practice on DFM →#436Inverse functions (excluding exponentials and modulus)
- Practice on DFM →#438Properties of functions
- Practice on DFM →#565One-to-one vs many-to-one functions
Quadratics
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- Practice on DFM →#366Plotting quadratic graphs from a table of values
- Practice on DFM →#368Quadratic graphs and their features
- Practice on DFM →#427Recognising the shape of more complex forms of quadratic, cubic, reciprocal and exponential graphs based on their equations
Exponential Functions
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- Practice on DFM →#423Plotting exponential graphs
- Practice on DFM →#424Properties of exponential graphs
- Practice on DFM →#425Determining the values of ||k|| and ||a|| in the exponential function ||y = ka^x||
Functions [EXT]
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- Practice on DFM →#435Composite functions (excluding exponentials and modulus)
- Practice on DFM →#570Sketching graphs involving the modulus function
Discriminant [EXT]
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- Practice on DFM →#492Discriminant of a quadratic function
Logs [EXT]
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- Practice on DFM →#394Fractional indices
- Practice on DFM →#527Laws of logs (excluding ||\ln(x)||)
- Practice on DFM →#529Graphs of ||y = \log_a(x)|| and ||y = \ln(x)||
Curated revision materials parents and students can download.
A review pack for this unit is being prepared.
MYP framing
- Focus ATLs
- Communication
- Self-management
- Thinking
- Key concepts
- Form
- Models
- Representations
- Global context
- Identities and Relationships