intro to Polynomial functions 

Functions vs relation

• Each X value gives only one Y value

  A vending machine

  1 button = 1 snack

Vertical line test ( how to check a graph )

• Draw an imaginary vertical line

• If the line touches once it’s a function

• If the line touches more than once, it’s not a function

Polynomial functions

• A polynomial is just a smooth curve made from powers of X

Examples

• X

• X²

• X³

• 2x - 3x + 1

Linear and quadratic functions are actually types of polynomial functions

Degree of a polynomial ( degree = highest exponent of X )

Examples

• X - degree 1

• X² - degree 2

• X³ + 4x - degree 3

The degree helps predict the graph shape

X-intercepts ( X-intercepts = where the graph crosses the X-axis )

At this point (y=0)

So find X-intercepts set f(X) = 0 and solve

These are also called the roots

End behaviour ( what the graph does on the far left and far right )

• Degree (odd or even)

• Sign of leading coefficient (+ or -)

The leading coefficient is the number in the front of the highest power of X

The 4 graph types

• Even degree + positive (both ends go up)

• Even degree + negative (both ends go down)

• Odd degree + positive (left down right up)

• Odd degree + negative (left up right down)

End behaviour depends on degree and the leading coefficient

Y-intercept (happens when X=0)

So plug in 0 to f(0)

Multiplicity (tells how the graph behaves when it hits the X-axis)

If factor appears

• (X-2)^1 crosses straight through

• (X-2)² touches and bounces

• (X-2)³ crosses but bends/flat

the order of the root controls how the graph crosses the intercept