# Functions and Algebra: Sketch and interpret functions and graphs

### Subject outcome

Subject outcome 2.1: Use a variety of techniques to sketch and interpret information from graphs of algebraic and transcendental functions.

### Learning outcomes

- Generate graphs by means of point-by-point plotting using/supported by available technology.
- Use the generated graphs to make and test conjectures.
- Investigate and generalise the impact of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] on the functions:
- [latex]\scriptsize y=ax+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}+q[/latex]
- [latex]\scriptsize y=\displaystyle \frac{a}{x}+q[/latex]
- [latex]\scriptsize y=a\cdot {{b}^{x}}+q[/latex] where [latex]\scriptsize b>0[/latex]
- [latex]\scriptsize y=\sin x+q[/latex]
- [latex]\scriptsize y=\cos x+q[/latex]
- [latex]\scriptsize y=\tan x+q[/latex]

- Define functions.
- Identify the following characteristics of functions:
- Domain and range
- Intercepts with axes
- Turning points, minima and maxima
- Asymptotes
- Shape and symmetry
- Periodicity and amplitude
- Functions or non-functions
- Continuous or discontinuous

- Sketch graphs and find equations of graphs for the following:
- [latex]\scriptsize y=ax+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}+q[/latex]
- [latex]\scriptsize y=\displaystyle \frac{a}{x}+q[/latex]
- [latex]\scriptsize y=a\cdot {{b}^{x}}+q[/latex] where [latex]\scriptsize b>0[/latex]
- [latex]\scriptsize y=\sin x+q[/latex]
- [latex]\scriptsize y=\cos x+q[/latex]
- [latex]\scriptsize y=\tan x+q[/latex]

### Unit 1 outcomes

By the end of this unit you will be able to:

- Define a function.
- Identify if a relationship is a function or not.
- Sketch and find the equation of a linear function ([latex]\scriptsize y=ax+q[/latex] or [latex]\scriptsize y=mx+c[/latex]).
- Explain the effects on the shape of the graphs of linear functions of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] or [latex]\scriptsize m[/latex] and [latex]\scriptsize c[/latex].
- Find the equation of a linear function from its graph or other details.
- State the domain and range of a linear function.

### Unit 2 outcomes

By the end of this unit you will be able to:

- Identify the following characteristics of quadratic functions:
- turning points
- minima and maxima
- shape and symmetry.

- Sketch and find the equation of the graph [latex]\scriptsize y=ax^2+q[/latex].
- Investigate and generalise the impact of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] on [latex]\scriptsize y=ax^2+q[/latex].

### Unit 3 outcomes

By the end of this unit you will be able to:

- Identify the following characteristics of functions:
- continuous or discontinuous
- asymptotes.

- Sketch and find the equation of the graph [latex]\scriptsize y=\displaystyle \frac{a}{x}+q[/latex].
- Investigate and generalise the impact of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] on [latex]\scriptsize y=\displaystyle \frac{a}{x}+q[/latex].

### Unit 4 outcomes

By the end of this unit you will be able to:

- Sketch and find the equation of the graph [latex]\scriptsize y=a.{{b}^{x}}+q,b>0[/latex].
- Investigate and generalise the impact of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] on [latex]\scriptsize y=a.{{b}^{x}}+q,b>0[/latex].

### Unit 5 outcomes

By the end of this unit you will be able to:

- Identify the following characteristics of trigonometric functions:
- periodicity
- amplitude.

- Sketch the graph [latex]\scriptsize y=a\sin x+q[/latex], [latex]\scriptsize y=a\cos x+q[/latex] and [latex]\scriptsize y=a\tan x+q[/latex].
- Identify the asymptotes of [latex]\scriptsize y=a\tan x+q[/latex].
- Investigate and generalise the impact of [latex]\scriptsize a[/latex] and [latex]\scriptsize q[/latex] on [latex]\scriptsize y=asinx+q[/latex], [latex]\scriptsize y=a\cos x+q[/latex] and [latex]\scriptsize y=a\tan x+q[/latex].