# Functions and algebra: Use a variety of techniques to sketch and interpret information from graphs of functions

### Subject outcome

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

### Learning outcomes

- Use a variety of techniques to sketch and interpret information from graphs of functions. (Sketching of graphs using point by point plotting is an option.)
- [latex]\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q[/latex]
- [latex]\scriptsize y=a{{x}^{2}}+bx+c[/latex]
- [latex]\scriptsize y=\displaystyle \frac{a}{{x+p}}+q[/latex]
- [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,b>0[/latex]
- [latex]\scriptsize y=a\sin kx[/latex]
- [latex]\scriptsize y=a\cos kx[/latex]
- [latex]\scriptsize y=a\tan kx[/latex]
- [latex]\scriptsize y=a\sin (x+p)[/latex]
- [latex]\scriptsize y=a\cos (x+p)[/latex]
- [latex]\scriptsize y=a\tan (x+p)[/latex]

**Note**: Cubic functions will only be done in differential Calculus in level 4.

- Investigate and generalize the impact of[latex]\scriptsize k,p,a,b,c[/latex] and [latex]\scriptsize q[/latex] on the functions listed above.
- 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
- Intervals in which a function increases/decreases.

- Find the equation of the graphs by calculations or using the method of inspection (investigating the transformation of the graph).

### Unit 1 outcomes

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

- Sketch quadratic functions of the form [latex]\scriptsize y=a{{x}^{2}}+bx+c[/latex] and [latex]\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q[/latex].
- Find the turning point of [latex]\scriptsize y=a{{x}^{2}}+bx+c[/latex] and [latex]\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q[/latex].

### Unit 2 outcomes

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

- Find the equation of a quadratic function when given the x-intercepts and another point on the graph.
- Find the equation of a quadratic function when given the turning point and another point on the graph.

### Unit 3 outcomes

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

- Determine the domain and range of hyperbolic functions of the form [latex]\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q[/latex].
- Find the asymptotes of hyperbolic functions of the form [latex]\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q[/latex].
- Find the axes of symmetry of hyperbolic functions of the form [latex]\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q[/latex].
- Find the intercepts with the axes of hyperbolic functions of the form [latex]\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q[/latex].
- Sketch hyperbolic functions of the form [latex]\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q[/latex].

### Unit 4 outcomes

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

- Find the equation of a hyperbola in the form [latex]\scriptsize y=\displaystyle \frac{a}{{x+p}}+q[/latex].

### Unit 5 outcomes

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

- Determine the domain and range of exponential functions of the form [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,b>0[/latex].
- Find the asymptote of exponential functions of the form [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,b>0[/latex].
- Find the intercepts with the axes of exponential functions of the form [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,b>0[/latex].
- Sketch exponential functions of the form [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,b>0[/latex].

### Unit 6 outcomes

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

- Find the equation of exponential graphs in the form [latex]\scriptsize y=a.{{b}^{{x+p}}}+q,\text{ }b>0[/latex].
- Analyse exponential functions.

### Unit 7 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=a\sin k\theta[/latex].
- Determine the effects of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] on the sine graph of the form [latex]\scriptsize y=a\sin k\theta[/latex].
- Find the values of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] from a given sine graph of the form [latex]\scriptsize y=a\sin k\theta[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\sin x[/latex] and [latex]\scriptsize y=\sin \theta[/latex] are the same function.

### Unit 8 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=\sin (\theta +p)[/latex].
- Determine the effects of positive and negative values of [latex]\scriptsize p[/latex] on the sine graph [latex]\scriptsize y=\sin (\theta +p)[/latex].
- Find the value of [latex]\scriptsize p[/latex] from a given sine graph of the form [latex]\scriptsize y=\sin (\theta +p)[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\sin x[/latex] and [latex]\scriptsize y=\sin \theta[/latex] are the same function.

### Unit 9 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=a\cos k\theta[/latex].
- Determine the effects of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] on the cosine graph of the form [latex]\scriptsize y=a\cos k\theta[/latex].
- Find the values of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] from a given cosine graph of the form [latex]\scriptsize y=a\cos k\theta[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\cos x[/latex] and [latex]\scriptsize y=\cos \theta[/latex] are the same function.

### Unit 10 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=\cos (\theta +p)[/latex].
- Determine the effects of positive and negative values of [latex]\scriptsize p[/latex] on the cosine graph [latex]\scriptsize y=\cos (\theta +p)[/latex].
- Find the value of [latex]\scriptsize p[/latex] from a given cosine graph of the form [latex]\scriptsize y=\cos (\theta +p)[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\cos x[/latex] and [latex]\scriptsize y=\cos \theta[/latex] are the same function.

### Unit 11 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=a\tan k\theta[/latex].
- Determine the effects of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] on the tangent graph of the form [latex]\scriptsize y=a\tan k\theta[/latex].
- Find the values of [latex]\scriptsize a[/latex] and [latex]\scriptsize k[/latex] from a given tangent graph of the form [latex]\scriptsize y=a\tan k\theta[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\tan x[/latex] and [latex]\scriptsize y=\tan \theta[/latex] are the same function.

### Unit 12 outcomes

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

- Sketch functions of the form [latex]\scriptsize y=\tan (\theta +p)[/latex].
- Determine the effects of positive and negative values of [latex]\scriptsize p[/latex] on the tangent graph [latex]\scriptsize y=\tan (\theta +p)[/latex].
- Find the value of [latex]\scriptsize p[/latex] from a given tangent graph of the form [latex]\scriptsize y=\tan (\theta +p)[/latex].

Remember that the domain of trigonometric functions can be represented as [latex]\scriptsize x[/latex] or [latex]\scriptsize \theta[/latex]. Therefore, [latex]\scriptsize y=\tan x[/latex] and [latex]\scriptsize y=\tan \theta[/latex] are the same function.