Before You Start Powers and exponents Factoring basics Reading intercepts from graphs What You Will Learn Distinguish power, polynomial, and rational functions. Use degree, leading coefficient, zeros, and factors to predict graph behavior. Identify vertical and horizontal asymptotes in simple rational functions. Preview Activity: Read the Shape Before the Formula A graph can reveal algebraic structure before you know the exact formula. 1. If a graph crosses the xaxis at x=2, x=1, and x=3, what factors might appear? 2. If both ends of the graph point upward, should the degree be even or odd? 3. If a graph touches the xaxis at x=1 but does not cross, what might be true about the factor x1? Hint: Crossing usually suggests an odd multiplicity. Touching and turning usually suggests an even multiplicity. Solution: Possible factors are (x+2), (x1), and (x3). Both ends in the same direction suggest even degree. Touching at x=1 suggests (x1) appears an even number of times. Power and Polynomial Functions p(x)=anx^n+a{n1}x^{n1}+\cdots+a1x+a0 A power function has the form f(x)=kx^n. A polynomial is a sum of power functions with nonnegative wholenumber exponents. The degree and leading coefficient usually control the farleft and farright behavior. Zeros and Turning Behavior The polynomial y=(x+2)(x1)^2(x3) crosses at x=2 and x=3, but touches at x=1. Zeros and Factors If p(a)=0, then a is a zero of the polynomial. A zero at x=a usually corresponds to a factor (xa). A polynomial of degree n has at most n real zeros and at most n1 turning points. Worked Example: Build a Polynomial from Zeros Build a polynomial with zeros 1, 2, and 2, and with positive leading coefficient. Solution: The zero 1 gives factor (x+1). The repeated zero 2 gives factor (x2)^2. One simple answer is p(x)=(x+1)(x2)^2. Rational Functions and Asymptotes r(x)=\frac{P(x)}{Q(x)} A rational function is a quotient of two polynomials. The denominator cannot be zero, so zeros of Q(x) are excluded from the domain. A vertical asymptote can occur where the denominator is zero and the factor does not cancel. Horizontal behavior depends on the degrees of P(x) and Q(x). Activity: Asymptote or Removable Hole? 1. Simplify r(x)=\frac{x^21}{x1} for x\ne 1. 2. Does x=1 create a vertical asymptote or a removable hole? 3. Compare with s(x)=\frac{x+1}{x1}. What happens near x=1? Hint: Factor x^21 first. Solution: x^21=(x1)(x+1), so r(x)=x+1 when x\ne 1. x=1 is a removable hole, not a vertical asymptote. For s(x) the factor does not cancel, so x=1 is a vertical asymptote. Summary Power functions are the building blocks; polynomials are sums of those blocks. Zeros and factors explain where a graph crosses or touches the xaxis. Rational functions require domain checks because division by zero controls holes and asymptotes. Try These 1. Predict the end behavior of p(x)=2x^5+3x^21. 2. Build a polynomial with zeros 0, 3, and 3. 3. Find the excluded values of r(x)=\frac{x+4}{x^29}.