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”. 🙂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)

hyperbolaslanted 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

 

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