# 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.)
• $\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q$
• $\scriptsize y=a{{x}^{2}}+bx+c$
• $\scriptsize y=\displaystyle \frac{a}{{x+p}}+q$
• $\scriptsize y=a.{{b}^{{x+p}}}+q,b>0$
• $\scriptsize y=a\sin kx$
• $\scriptsize y=a\cos kx$
• $\scriptsize y=a\tan kx$
• $\scriptsize y=a\sin (x+p)$
• $\scriptsize y=a\cos (x+p)$
• $\scriptsize y=a\tan (x+p)$
Note: Cubic functions will only be done in differential Calculus in level 4.
• Investigate and generalize the impact of$\scriptsize k,p,a,b,c$ and $\scriptsize q$ 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 $\scriptsize y=a{{x}^{2}}+bx+c$ and $\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q$.
• Find the turning point of $\scriptsize y=a{{x}^{2}}+bx+c$ and $\scriptsize \displaystyle y=a{{(x+p)}^{2}}+q$.

### 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 $\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q$.
• Find the asymptotes of hyperbolic functions of the form $\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q$.
• Find the axes of symmetry of hyperbolic functions of the form $\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q$.
• Find the intercepts with the axes of hyperbolic functions of the form $\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q$.
• Sketch hyperbolic functions of the form $\scriptsize y=\displaystyle \frac{a}{{(x+p)}}+q$.

### Unit 4 outcomes

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

• Find the equation of a hyperbola in the form $\scriptsize y=\displaystyle \frac{a}{{x+p}}+q$.

### 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 $\scriptsize y=a.{{b}^{{x+p}}}+q,b>0$.
• Find the asymptote of exponential functions of the form $\scriptsize y=a.{{b}^{{x+p}}}+q,b>0$.
• Find the intercepts with the axes of exponential functions of the form $\scriptsize y=a.{{b}^{{x+p}}}+q,b>0$.
• Sketch exponential functions of the form $\scriptsize y=a.{{b}^{{x+p}}}+q,b>0$.

### Unit 6 outcomes

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

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

### Unit 7 outcomes

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

• Sketch functions of the form $\scriptsize y=a\sin k\theta$.
• Determine the effects of $\scriptsize a$ and $\scriptsize k$ on the sine graph of the form $\scriptsize y=a\sin k\theta$.
• Find the values of $\scriptsize a$ and $\scriptsize k$ from a given sine graph of the form $\scriptsize y=a\sin k\theta$.

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

### Unit 8 outcomes

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

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

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

### Unit 9 outcomes

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

• Sketch functions of the form $\scriptsize y=a\cos k\theta$.
• Determine the effects of $\scriptsize a$ and $\scriptsize k$ on the cosine graph of the form $\scriptsize y=a\cos k\theta$.
• Find the values of $\scriptsize a$ and $\scriptsize k$ from a given cosine graph of the form $\scriptsize y=a\cos k\theta$.

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

### Unit 10 outcomes

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

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

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

### Unit 11 outcomes

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

• Sketch functions of the form $\scriptsize y=a\tan k\theta$.
• Determine the effects of $\scriptsize a$ and $\scriptsize k$ on the tangent graph of the form $\scriptsize y=a\tan k\theta$.
• Find the values of $\scriptsize a$ and $\scriptsize k$ from a given tangent graph of the form $\scriptsize y=a\tan k\theta$.

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

### Unit 12 outcomes

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

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

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