Quadratic Functions In Chapter 3, we will discuss polynomial functions Def. A polynomial is a function that adds integer powers of x, each of which has a constant coefficient. The highest power of x is called the degree of the polynomial. There are no roots and we don't divide by x.

Remember, the zeros of a function are the values of x that make f (x) = 0 Ex. Find the zeros of f (x) = -2x 4 + 2x 2

A polynomial of degree 2 is called a quadratic function. The general form of a quadratic function is The graph is called a parabola.

Parabolas are symmetric with respect to a vertical line, called the axis of symmetry This line has the equation The turning point of a parabola is called the vertex. Notice that the x-coordinate of the vertex is also If the lead coefficient, a, is positive, the parabola opens upward.

If the lead coefficient, a, is negative, the parabola opens downward.

Ex. A baseball is hit so that the path of the ball is given by the function f (x) = x 2 + x + 3, where f (x) is the height of the ball (in ft.) and x is the horizontal distance from home plate (in ft.). What is the maximum height reached by the baseball?

Ex. Sketch the graphs of y = x 2, y = 2x 2, and y = x 2

The standard form of a quadratic function is The vertex of the parabola is the point (h,k). If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

Ex. Sketch the graph of f (x) = 2x 2 + 8x + 7

Ex. Find the vertex and x-intercepts of f (x) = -x 2 + 6x – 8, then use them to sketch the graph.

Ex. Write the equation of the parabola whose vertex is (1,2) and that contains the point (0,0).

Practice Problems Section 3.1 Problems 13, 23, 43, 77

Polynomial Functions While graphing higher order polynomials will be difficult, there are some things that will help. All polynomials are continuous (no gaps or jumps). polynomialnot a polynomial

All polynomials are smooth (no sharp turns). polynomialnot a polynomial

Some simple polynomials to graph look like f (x) = x n, where n is a positive integer. These are called power functions. y = x 2 y = x 3 y = x 5 y = x 4 y = x 6 y = x 7

Ex. Sketch f (x) = (x + 1) 4

The lead coefficient can tell you about the “end behavior” of the graph - What happens if the graph keeps going left or right - The degree of the polynomial affects the results

If the degree is odd: Positive lead coeff. Negative lead coeff.

If the degree is even: Positive lead coeff. Negative lead coeff.

Ex. Find the degree of the polynomial and describe the end behavior of the graph. a) f (x) = -x 3 + 4x b) f (x) = -5x + x 4 + 4

Remember, the zeros of a function are the values of x that make f (x) = 0 If the degree of a polynomial is n, then there will be at most n zeros on the graph

Ex. Find the zeros of f (x) = -2x 4 + 2x 2 Let's look at the graph of the function.

Ex. Sketch the graph of f (x) = 3x 4 – 4x 3

Intermediate Value Theorem If f is a polynomial then, on the interval [a,b], f takes on every value between f (a) and f (b).

Ex. Use the Intermediate Value Theorem to approximate the zero of f (x) = x 3 – x 2 + 1

Practice Problems Section 3.2 Problems 9b, 15, 27, 67, 85

Polynomial Division Ex.

Ex. Divide 6x 3 – 19x x – 4 by x – 2, and use the results to factor the polynomial completely.

Ex. Divide x 2 + 3x + 5 by x + 1.

Ex. Divide x 3 – 1 by x – 1.

Ex. Divide 2x 4 + 4x 3 – 5x 2 + 3x – 2 by x 2 + 2x – 3.

Ex. Divide x 4 – 10x 2 – 2x + 4 by x + 3.

Ex. Show that (x – 2) and (x + 3) are factors of f (x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18, then find the remaining factors.

Practice Problems Section 3.3 Problems 5, 13, 15, 19, 57

Zeros of a Polynomial Fundamental Theorem of Algebra If a polynomial has degree n, then it has exactly n zeros. - This includes repeating zeros - Some of the zeros may be complex numbers - If k is a zero, then the polynomial can be divided by (x - k) - We know that the zeros exist, though we may not be able to find them

a) f (x) = x – 2 has one zero, x = 2 b) f (x) = x 2 – 6x + 9 has two zeros, x = 3 and x = 3 c) f (x) = x 3 + 4x has three zeros, x = 0, x = 2i, and x = -2i d) f (x) = x 4 – 1 has four zeros, x = 1, x = -1, x = i, and x = -i

To help us find zeros, we can use: Rational Zero Test If a polynomial has integer coefficients, every rational zero has the form where p = factor of constant coefficient q = factor of lead coefficient

Ex. Find the rational zeros of f (x) = x 3 + x + 1

Ex. Find the rational zeros of f (x) = x 4 – x 3 + x 2 – 3x – 6, then factor.

Ex. Find all real solutions of 2x 3 + 3x 2 – 8x + 3 = 0

If a complex number is a zero of a polynomial, then so is the conjugate. Ex. Find a fourth-degree polynomial that has 1, -1, and 3i as zeros.

Ex. Find all zeros of f (x) = x 4 – 3x 3 + 6x 2 + 2x – 60 given that 1 + 3i is a zero.

Ex. Express f (x) = x 4 – 3x 3 + 6x 2 + 2x – 60 as the product of linear factors.

Practice Problems Section 3.4 Problems 1, 7, 11, 17, 23, 37, 47

Mathematical Modeling In the real world, you will have data points and will want to find a function to describe the situation.

Ex. Use your calculator to find a linear regression for the data below.

y is directly proportional to x if y = kx for some constant k k is called the constant of proportionality We also say varies directly

Ex. The state income tax is directly proportional to gross income. If the tax is $46.05 for an income of $1500, write a mathematical model for income tax.

y is directly proportional to the n th power to x if y = kx n for some constant k Ex. The distance a ball rolls down a hill is directly proportional to the square of the time it rolls. During the first second, the ball rolls 8 ft. Write an equation for this model. How long does it take for the ball to roll 72 ft?

y is inversely proportional to x if for some constant k We also say varies inversely z is jointly proportional to x and y if for some constant k We also say varies jointly

Ex. The simple interest for a savings account is jointly proportional to the time and the principal. After one quarter, the interest on a principal of $5000 is $ Write an equation for this model.

Practice Problems Section 3.5 Problems 7, 33, 63, 65, 67