function – Mathtuition88

Domain and Range of Inverse Helpsheet

Here is a helpsheet that I created on the topic of Domain and Range of Inverse Functions, a topic no longer in O Levels but still in IP Math.

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Functions and Graphs

In this post, we will discuss how to sketch the graphs of $y=ax^n$ , for y=-2 to 3.

First, we will look at Quadratic Graphs ( $y=ax^2+bx+c$ )

Quadratic Graphs

If a>0, the graph is a “U” shape or “happy face”. 🙂

If a<0, the graph is a “n” shape or “sad face”. 😦

sad face

Intercepts

To find x-intercept: Let y=0
To find y-intercept: Let x=0

Completing the square

E.g. $2x^2+8x+4=2(x^2+4x+2)$ (take out common factor of $x^2$ )
= $2(x^2+4x \mathbf{+2^2-2^2}+2)$ (Key step: Divide coefficient of x by 2, add & subtract the square of it)
Check your answer using substitution method! (Sub. in $x=9$ into your initial and final answer)

Turning Points

$y=\pm (x-p)^2+q$

* (p,q) is the turning point of the graph

For example, $y=(x-2)^2+3$

minimum point

Minimum pt: (2,3)

Graphs of $y=ax^n$

$\displaystyle y=ax^{-2}=\frac{a}{x^2}$ , where a>0 (Volcano shape)

1 over x2
Not touching the x-axis (Asymptote)
volcano

If a<0, the graph becomes upside down.

minus 1 over x2

( $\displaystyle y=\frac{-1}{x^2}$ )

$\displaystyle y=ax^{-1}=\frac{a}{x}$ , where a>0. (Hyperbola, Slanted Hourglass)

hyperbola slanted hourglass

If a<0, it will be upside down (reflected about x-axis)

minus 1 over x ( $\displaystyle y=\frac{-1}{x}$ )

Cubic $y=x^3$

cubic