Integrated Math 3 / Unit 1: Polynomial Functions
1.6 Graphing and Modeling with Polynomials
Notes
To sketch a polynomial: find the zeros and their multiplicities, the $y$-intercept $f(0)$, and the end behavior. Then connect the pieces.
- Write a polynomial from zeros: $f(x) = a(x - r_1)(x - r_2)\cdots$, then use a point to find $a$.
- The Intermediate Value Theorem: if $f(a)$ and $f(b)$ have opposite signs, there is a zero between $a$ and $b$.
- Polynomials are used to model volume. An open box made from a 12 inch by 12 inch sheet by cutting squares of side $x$ from the corners has $V = x(12 - 2x)^2$, with $0 < x < 6$.
Practice
A quadratic $f$ has zeros at $-1$ and $4$ and passes through $(0, 8)$. What is $f(x)$?
$f(0) = a(1)(-4) = 8$ gives $a = -2$.
For a continuous function $f$, $f(1) = -3$ and $f(2) = 5$. What must be true?
The values change sign, so by the Intermediate Value Theorem the graph crosses the $x$-axis in between. It need not be at 1.5.
An open box is made from a 12 inch square sheet by cutting $2$ inch squares from the corners. What is its volume?
$V = x(12 - 2x)^2 = 2(8)^2 = 128$.
Ready for more? Take the Unit 1 test.