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 $\scriptsize a$ and $\scriptsize q$ on the functions:
• $\scriptsize y=ax+q$
• $\scriptsize y=a{{x}^{2}}+q$
• $\scriptsize y=\displaystyle \frac{a}{x}+q$
• $\scriptsize y=a\cdot {{b}^{x}}+q$ where $\scriptsize b>0$
• $\scriptsize y=\sin x+q$
• $\scriptsize y=\cos x+q$
• $\scriptsize y=\tan x+q$
• 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:
• $\scriptsize y=ax+q$
• $\scriptsize y=a{{x}^{2}}+q$
• $\scriptsize y=\displaystyle \frac{a}{x}+q$
• $\scriptsize y=a\cdot {{b}^{x}}+q$ where $\scriptsize b>0$
• $\scriptsize y=\sin x+q$
• $\scriptsize y=\cos x+q$
• $\scriptsize y=\tan x+q$

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 ($\scriptsize y=ax+q$ or $\scriptsize y=mx+c$).
• Explain the effects on the shape of the graphs of linear functions of $\scriptsize a$ and $\scriptsize q$ or $\scriptsize m$ and $\scriptsize c$.
• 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 $\scriptsize y=ax^2+q$.
• Investigate and generalise the impact of $\scriptsize a$ and $\scriptsize q$ on $\scriptsize y=ax^2+q$.

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 $\scriptsize y=\displaystyle \frac{a}{x}+q$.
• Investigate and generalise the impact of $\scriptsize a$ and $\scriptsize q$ on $\scriptsize y=\displaystyle \frac{a}{x}+q$.

Unit 4 outcomes

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

• Sketch and find the equation of the graph $\scriptsize y=a.{{b}^{x}}+q,b>0$.
• Investigate and generalise the impact of $\scriptsize a$ and $\scriptsize q$ on $\scriptsize y=a.{{b}^{x}}+q,b>0$.

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 $\scriptsize y=a\sin x+q$, $\scriptsize y=a\cos x+q$ and $\scriptsize y=a\tan x+q$.
• Identify the asymptotes of $\scriptsize y=a\tan x+q$.
• Investigate and generalise the impact of $\scriptsize a$ and $\scriptsize q$ on $\scriptsize y=asinx+q$, $\scriptsize y=a\cos x+q$ and $\scriptsize y=a\tan x+q$.