8/1/015 The Graph of a Quadratic Function Quadratic Functions & Models Precalculus.1 The Graph of a Quadratic Function The Graph of a Quadratic Function All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola, as shown in Figure.1. The graph of a quadratic function is a special type of U -shaped curve called a parabola. Parabolas occur in many real-life applications especially those involving reflective properties of satellite dishes and flashlight reflectors. Leading coefficient is positive. Leading coefficient is negative. Figure.1 The Graph of a Quadratic Function If the leading coefficient is positive, the graph of f(x) = ax + bx + c is a parabola that opens upward. If the leading coefficient is negative, the graph of f(x) = ax + bx + c is a parabola that opens downward. The simplest type of quadratic function is f(x) = ax. The Graph of a Quadratic Function Its graph is a parabola whose vertex is (0, 0). If a > 0, the vertex is the point with the minimum y-value on the graph, and if a < 0, the vertex is the point with the maximum y-value on the graph, as shown in Figure.. Leading coefficient is positive. Figure. Leading coefficient is negative. 1
8/1/015 Example 1 Sketching Graphs of Quadratic Functions a. Compare the graphs of y = x and f(x) = x. b. Compare the graphs of y = x and g(x) = x. Solution: a. Compared with y = x, each output of f(x) = x shrinks by a factor of, creating the broader parabola shown in Figure.. Example 1 Solution b. Compared with y = x, each output of g(x) = x stretches by a factor of, creating the narrower parabola shown in Figure.4. cont d Figure. Figure.4 The Standard Form of a Quadratic Function Example 1 Sketch and compare the graphs of and 1 g( x) x to x. 6 h( x) x 5 Example Sketch the graph and identify the vertex and the axis of the parabola. x 10x 5 Example Sketch the graph and identify the vertex and the axis of the parabola. x 4x 1
8/1/015 Example 4 Finding Minimum and Maximum Values Write the standard form of the equation of the parabola whose vertex is (-4,11) and that passes through the point (-6,15). What would the equation be if the vertex was still (-4,11) and it passed through the point (,0)? Example 5 The height y (in feet) of a ball thrown by a child is given by 1 x x 4 8 Where x is the horizontal distance (in feet) from where the ball is thrown. How high is the ball when it is at its maximum height? Polynomial Functions of Higher Degree Precalculus. Graphs of Polynomial Functions In this section, we will study basic features of the graphs of polynomial functions. The first feature is that the graph of a polynomial function is continuous. Graphs of Polynomial Functions The graph shown in Figure.11(b) is an example of a piecewise defined function that is not continuous. Essentially, this means that the graph of a polynomial function has no breaks, holes, or gaps, as shown in Figure.11(a). Polynomial functions have continuous graphs. Figure.11(a) Functions with graphs that are not continuous are not polynomial functions. Figure.11(b)
8/1/015 Graphs of Polynomial Functions From Figure.14, you can see that when n is even, the graph is similar to the graph of f (x) = x, and when n is odd, the graph is similar to the graph of f (x) = x. Graphs of Polynomial Functions Moreover, the greater the value of n, the flatter the graph near the origin. Polynomial functions of the form f (x) = x n are often referred to as power functions. If n is even, the graph of y = x n touches the axis at the x-intercept. (a) If n is old, the graph of y = x n crosses the axis at the x-intercept. (b) Figure.14 Example 1 The Leading Coefficient Test Sketch the graph of each function: a) ( x) x 5 b) k( x) x f 4 1 4 4 The Leading Coefficient Test cont d Example Describe the left-hand and right-hand behavior of the graph of each function. 1 5 a) x x b).6x 5x 1 4 4
8/1/015 Zeros of Polynomial Functions Example Find all the real zeros of x 1x 6x. Then determine the number of turning points of the graph of the function. Zeros of Polynomial Functions Example 4 Sketch the graph of x 6x. Example 5 4 9 Sketch the graph of 1 x x x. 4 4 The Intermediate Value Theorem The next theorem, called the Intermediate Value Theorem, illustrates the existence of real zeros of polynomial functions. This theorem implies that if (a, f(a)) and (b, f(b)) are two points on the graph of a polynomial function such that f(a) f(b), then for any number d between f(a) and f(b) there must be a number c between a and b such that f(c) = d. (See Figure.5.) Figure.5 5
8/1/015 The Intermediate Value Theorem The Intermediate Value Theorem helps you locate the real zeros of a polynomial function in the following way. If you can find a value x = a at which a polynomial function is positive, and another value x = b at which it is negative, you can conclude that the function has at least one real zero between these two values. Example 6 Use the Intermediate Value Theorem to approximate the real zeros of the function to the nearest hundredth. 5 x x x Example 1 Polynomial & Synthetic Division Divide x x 9 by x using long division. Use the result to factor the polynomial completely. Precalculus. Example Long Division of Polynomials Divide x 4 1 by x 1. 6
8/1/015 Example 4 Divide 6x x x 9x by x x 1. Synthetic Division There is a nice shortcut for long division of polynomials by divisors of the form x k. This shortcut is called synthetic division. Example 4 Use synthetic division to divide 5x by x. 8x x 6 Don t forget place holders! Ex: Synthetic division of x 4 1 by x 1. The Remainder and Factor Theorems The Remainder Theorem tells you that synthetic division can be used to evaluate a polynomial function. That is, to evaluate a polynomial function f (x) when x = k, divide f (x) by x k. The remainder will be f (k), as illustrated in Example 5. Example 5 Use the Remainder Theorem to evaluate the function at f(-1). 4x 10x x 8 7
8/1/015 The Remainder and Factor Theorems Example 6 Show that (x+) is a factor of x 19x 0. Then find the remaining factors of f(x). The Remainder and Factor Theorems Complex Numbers Precalculus.4 The Imaginary Unit i The Imaginary Unit i You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation x + 1 = 0 has no real solution because there is no real number x that can be squared to produce 1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit i, defined as In the standard form a + bi, the real number a is called the real part of the complex number a + bi, and the number bi (where b is a real number) is called the imaginary part of the complex number. i = Imaginary unit where i = 1. 8
8/1/015 The Imaginary Unit i Write the following in standard form: 5 9 Operations with Complex Numbers Operations with Complex Numbers The additive identity in the complex number system is zero (the same as in the real number system). Furthermore, the additive inverse of the complex number a + bi is (a + bi) = a bi. So, you have (a + bi) + ( a bi) = 0 + 0i = 0. Additive inverse Example 1 Perform the addition or subtraction and write the result in standard form. a) ( 7 i) (5 4i) b) ( 4i) (5 i) c) 4i ( 5i) ( 6i) d) ( 5 i) ( 4i) (8 i) Operations with Complex Numbers Many of the properties of real numbers are valid for complex numbers as well. Here are some examples: Associative Properties of Addition and Multiplication Commutative Properties of Addition and Multiplication Distributive Property of Multiplication Over Addition 9
8/1/015 Example Perform the multiplication and write the result in standard form. a) 8i(4 i) b) ( i)(5 4i) c) ( 4i)( 4i) d) ( 5 i) Complex Conjugates The product of two complex numbers can be a real number. This occurs with pairs of complex numbers of the form a + bi and a bi, called complex conjugates. (a + bi)(a bi) = a abi + abi b i = a b ( 1) = a + b Example Multiply each complex number by its complex conjugate. a) i Example 4 Write the quotient in standard form. i i b) 5 4i Complex Solutions of Quadratic Equations Example 5 Write the complex number in standard form. a) 14 b) c) 7 1 5 10
8/1/015 Example 6 Use the Quadratic Formula to solve the quadratic equations. a) 8x 14x 9 0 b) 7x 5x 0 Review Quiz.1 to.4 Parabolas- standard form, vertex, axis symmetry Complete the square Left/Right hand behavior Find zeros Long/ synthetic division Complex number in standard form The Fundamental Theorem of Algebra Zeros of Polynomial Functions Precalculus.5 Using the Fundamental Theorem of Algebra and the equivalence of zeros and factors, you obtain the Linear Factorization Theorem. Example 1 The Rational Zero Test Find the zeros of each function. a) x 4 b) x 4x 4 c) x 9x d) x 4 16 11
8/1/015 Example Find the rational zeros of the function: x 5x x 8 Example Find the rational zeros of the function: x 15x 75x 15 Example 4 Find the rational zeros of the function: x 4 9x 18x 71x 0 Example 5 Find all the real solutions of the function: x 7x 11x 14 0 Conjugate Pairs Example 6 Find a rd degree polynomial with integer coefficients that has and 7i as zeros. 1
8/1/015 Factoring a Polynomial The following theorem says that even if you do not want to get involved with complex factors, you can still write f (x) as the product of linear and/or quadratic factors. Example 7 Find all the zeros of the function, given that (1+4i) is a zero of f. x 4x 1x 4 Example 8 Other Tests for Zeros of Polynomials Write the function as a product of linear factors, and list all of its zeros. h( x) x 11x 41x 51 A variation in sign means that two consecutive coefficients have opposite signs. Example 9 Describe the possible zeros of the function: x 5x x 8 Rational Functions Precalculus.6 1
8/1/015 Introduction A rational function is a quotient of polynomial functions. It can be written in the form Example 1 Find the domain of the function and discuss the behavior of f near any excluded x-values. x x 1 where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. Vertical and Horizontal Asymptotes Example Find the horizontal and vertical asymptotes of the function. 5x x 1 Vertical and Horizontal Asymptotes Example Sketch the graph of the function and state it s domain. 1 x 14
8/1/015 Example 4 Analyzing Graphs of Rational Functions Sketch the graph of the function and state it s domain. x 1 x Example 5 Sketch the graph of the function. x x x Example 6 Sketch the graph of the function. x 4 x 4x 4 Slant Asymptotes Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. Slant Asymptotes To find the equation of a slant asymptote, use long division. For instance, by dividing x + 1 into x x, you obtain For example, the graph of has a slant asymptote, as shown in Figure.45. As x increases or decreases without bound, the remainder term /(x + 1) approaches 0, so the graph of f approaches the line y = x as shown in Figure.45. Figure.45 15
8/1/015 Example 7 Sketch the graph of the function. x 1 x Example 8 The cost C (in millions of dollars) for removing p% of the industrial pollutants discharged into a river is given by 55p C,0 p 100 100 p A proposed new law would require companies to remove 80% of their pollutants discharged into the river. The current law requires 45% removal. How much additional cost would the companies incur as a result of this law? Example 9 A rectangular page is designed to contain 57 square inches of print. The margins on each side, as well as the top and bottom, are all 1 inch deep. What should the dimensions of the page be so that the least amount of paper is used? Nonlinear Inequalities Precalculus.7 Polynomial Inequalities These zeros are the key numbers of the inequality, and the resulting intervals are the test intervals for the inequality. For instance, the polynomial above factors as Polynomial Inequalities These zeros divide the real number line into three test intervals: (, 1), ( 1, ), and (, ). (See Figure.5.) x x = (x + 1)(x ) and has two zeros, x = 1 and x =. Three test intervals for x x Figure.5 16
8/1/015 Polynomial Inequalities Example 1 Solve x 5x 6 0 Example Solve x 4x 1x 16 Solve x 11x 4 Example Example 4 Example 5 Solve the inequalities: a) x 6x 1 0 Solve x 5 1 x b) c) d) x 4x 4 0 x 5x 7 0 x 6x 9 0 17
8/1/015 Example 6 The marketing department of a manufacturer of personal electronic organizers has determined that the demand for a new model is p 500 0. 00005x Where p is the price per organizer and x is the number of organizers sold. The revenue for selling x organizers is R x( 500 0.00005x) The total cost is C 50x,500,000 What price should the company charge per organizer to obtain a profit of at least $960,000,000? Find the domain of Example 7 16 6x 18