# 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 ( $y=ax^2+bx+c$)

If a>0, the graph is a “U” shape or “happy face”. 🙂 If a<0, the graph is a “n” shape or “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 pt: (2,3)

## Graphs of $y=ax^n$

### $\displaystyle y=ax^{-2}=\frac{a}{x^2}$, where a>0 (Volcano shape) Not touching the x-axis (Asymptote) If a<0, the graph becomes upside down. ( $\displaystyle y=\frac{-1}{x^2}$)

### $\displaystyle y=ax^{-1}=\frac{a}{x}$, where a>0. (Hyperbola, Slanted Hourglass)  If a<0, it will be upside down (reflected about x-axis) ( $\displaystyle y=\frac{-1}{x}$)

### Cubic $y=x^3$  ## Author: mathtuition88

https://mathtuition88.com/

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