2.1 Quadratic Functions

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1 Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function. Quadratic Function: a polynomial function of degree Graph is u-shaped, called a parabola Parabolas are symmetric with respect to a line called the axis of symmetry Parabolas have a vertex, which is the point on the parabola where it intersects the axis of symmetry, and is the maximum or minimum of the quadratic function General Form: b f x ax bx c Axis of Symmetry: x a o o o x-intercepts can be found using the quadratic formula: x-intercepts can be found by factoring. x-intercepts can be found by completing the square. Vertex Form (standard form): f x a x h k Vertex: hk, Axis of Symmetry: x Graphing a Parabola h x b b 4ac o In both forms, if a 0, the parabola opens UP; if a 0, the parabola opens DOWN Ex1a: Write the equation in vertex form. Find the vertex and axis of symmetry. Then graph the f x x 4x 1 function. a Ex. 1b. You Try: Write in vertex form. Find vertex & axis of symmetry: f(x) = -6x + x 3 f x x 6x 8 Ex. 1c. Identify the x-intercepts of the quadratic function Page 1 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

2 Precalculus Notes: Unit Polynomial Functions Writing the Equation of a Parabola f ( x) a( x h) k Substitute the vertex for hk, in vertex form. Substitute the given point for xy, and solve for a. Exa: Write an equation for the parabola with vertex hk, xy, Ex.b You try: Write the equation for a parabola with a vertex (1,) that passes through the point (0, 5). 3, 4 that passes through the point 4, 6. Average Rate of Change (recall): f b b a f a Ex3: Find the average rate of change of f x x from x to x 4. Ave Rate of Change = Vertical Free-Fall Motion: formula for the position of an object in free fall 1 st gt v0t s0 g 3 ft/sec 9.8 m/sec (on Earth) s = position g = acceleration due to gravity v 0 = initial velocity s 0 = initial position Ex4: A flare is shot straight up from a ship s bridge 75 feet above the water with an initial velocity of 76 ft/sec. How long does it take to reach its maximum height? How long does it take to reach 163 feet? The flare reaches it max height at the vertex of the parabola. It takes sec for the flare to reach its maximum height. Note: This is NOT the graph of the path of the flare. The graph relates the position of the flare to time. The flare reaches 163 feet when t 76t 75. Algebraic Method: It reaches 163 feet at Page of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

3 Precalculus Notes: Unit Polynomial Functions Ex. 5 A soda pop company s sales average 6,000 cans per month when the cans sell for 50 cents each. For each nickel increase in the price, the sales per month drop by 1000 cans. Find the maximum revenue. You Try: Write the function in vertex form. Find the vertex and axis of symmetry. Then graph. f x x x 3 Reflection: Using a table of values, how can you determine whether it is a linear or quadratic function? Page 3 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

4 Date:. Polynomial Functions of Higher Degree Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph of a polynomial, radical, or rational function. Polynomial Function: a function that can be written in the form n n1... f x a x a x a x a x a, where n is a nonnegative integer and a 0 n n1 1 0 n Note: This is called Standard Form when the exponents of x descend Degree: the largest exponent, n, of x Leading Coefficient: the coefficient of the first term when written in standard form Constant Term: the numerical term, a 0 The graph of a polynomial function is a continuous curve with smooth round turns. Classifying Polynomial Functions I. Number of Terms 1 term = monomial terms = binomial 3 terms = trinomial 4 or more terms = polynomial II. Degree Degree Name (Degree) Standard Form Example Classification of Example 0 Constant f x a y 3 1 Linear f x ax b Quadratic f x ax bx c 3 Cubic 3 f x ax bx cx d 4 Quartic 4 3 f x ax bx cx dx e 5 Quintic f x ax bx cx dx ex f Constant Monomial y x Linear Monomial y x x 5 7 Quadratic Trinomial y 3 5x Cubic 4 y x x x Binomial 9 1 Quartic Polynomial y 5 x Quintic Monomial Page 4 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

5 Precalculus Notes: Unit Polynomial Functions Ex1: Which of the following are polynomial functions? State the degree and the leading coefficient or explain why it is not a function. g x 3x 17 a) 8 h x 13x 7x 11 b) 5 Ex Describe how to transform the graph. Name the y-intercept. f x x 4 7 a.) 3 b.) g(x) = (6 3x) 4 1 The Leading Coefficient Test to Predict End Behavior of Polynomial Functions Odd Degree: If the leading coefficient is positive, then lim & lim. x x If the leading coefficient is negative, then lim & lim. x x Even Degree: If the leading coefficient is positive, then lim & lim. x x LIf the leading coefficient is negative, then lim & lim. x x Ex: Indicate if the degree of the polynomial function shown in the graph is odd or even and indicate the sign of the leading coefficient. (see pg. 105 exploration) a.) b.) Page 5 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

6 Precalculus Notes: Unit Polynomial Functions Zeros (x-intercepts, Roots) of Polynomial Functions: nth-degree polynomials have at most n 1 extrema (relative minima or maxima) and n zeros. Recall that a zero of a function f is a number x for which f(x)=0. 1. x=a is a zero of function f.. x=a is solution of the polynomial function f(x)=0 3. (x-a) is a factor of the polynomial f(x). 4. (a,0) is an x-intercept of the graph of f. Zeros of Polynomial Functions Multiplicity: repeated zeros; If a polynomial function f has a factor of x then c is a zero of multiplicity m of f. Odd Multiplicity: f crosses the x-axis at c; f x changes signs m m 1 c, and not x c, Even Multiplicity: f bounces or is tangent to the x-axis at c; f x doesn t change signs 3 Ex. 3 Sketch the graph of g x x x 1. Describe the multiplicity of the zeros. End behavior (degree of, odd or even, leading coefficient): ; lim x lim x Domain: Continuity: Symmetry: Range: y Boundedness: Local Extrema: Asymptotes: x Page 6 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

7 Precalculus Notes: Unit Polynomial Functions f x x x Ex5: Graph the function and list its characteristics. 3 Domain: Range: Increasing: Decreasing: Symmetry: Boundedness: Extrema: Local Max: Local Min: End Behavior: lim f( x) ; lim f( x) Zeros: x x 1 1 f x x 5x 4. Graph: Ex6: Find the zeros and extrema of the function 4 Zeros: Extrema: y x Newton s Law of Cooling Ex7: The rate at which an object cools varies as the difference between its temperature and the temperature of the surrounding air. When a 70 C steel plate is placed in air that is at 0 C, it is cooling at 50 C per minute. How fast is it cooling when its temperature is 100 C? R = rate of cooling, T 0 = initial temp, S Solve for k first: T = surrounding temp: R k T T 0 S Page 7 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

8 Precalculus Notes: Unit Polynomial Functions Watch for hidden behavior: Ex8: Graph the function in a Standard window and find the zeros. 3 f x x 31x 8x 60 f x is degree 3, so the end behavior is lim, so we know it must cross the x-axis again to the right. x New window: Zeros are Intermediate Value Theorem (Location Principle): If a and b are real numbers with a b, and if f is continuous on ab,, then f takes on every value between f a and f b. Therefore, if f a and f b have opposite signs, then f c 0 for some number c in, ab. Ex9: Use the Intermediate Value Theorem to show 3 in these intervals: 1,0 and 1, f changes sign (negative to positive) in the interval f x 1x 55x 18x 40 has zeros 1,0, so f must have at least one zero in 1,0. f changes sign (positive to negative) in the interval 1,, so f must have at least one zero in 1,. Application of Polynomial Functions Ex10: You cut equal squares from the corners of a by 30 inch sheet of cardboard to make a box with no top. What size squares would need to be cut for the volume to be 300 cubic inches? Define x as the length of the sides of the squares. Write a formula for the volume of the box. Vx Solve the equation by graphing. Squares with lengths of approximately inches should be cut. Page 8 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

9 Precalculus Notes: Unit Polynomial Functions Ex. 11 Find polynomial functions with the following zeros. (There are many correct solutions.) a.) 1,3,3 b.) 3, 11, 11 You Try: Sketch the graph of the polynomial function. g(x) = (x-)3 (x + 1) f(x) =.(x +1) (x 3)(x + 5) Reflection: How many zeros can a function of degree n have? Explain your answer. Page 9 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

10 Date:.3 Real Zeros of Polynomial Functions Precalculus Notes: Unit Polynomial Functions Syllabus Objectives:. The student will calculate the intercepts of the graph of a given relation. Review: Long Division Long Division of Polynomials (same process!) Ex1a: Find the quotient. x 4 8x 3 11x 6 x 3 Note: Every term of the polynomial in the dividend must be represented. Since this polynomial is missing an x term, we must include the term 0x. x x x x x Solution: Remainder Theorem: If f x is divided by x k, then the remainder, r f k. Exa: Find the remainder without using division for 4 3 f x x 8x 11x 6 r f x x x Note: Compare your answer to the long division problem above divided by x 3. Page 10 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

11 Precalculus Notes: Unit Polynomial Functions Ex. b. You try: (x 3 3x 5x + 6) (x 3) Factor Theorem: x k is a factor of f x if and only if 0 f k. Ex3: Determine if x 3 is a factor of Show that r f 3 0: so by the Factor Theorem, x 3 is a factor of Synthetic Substitution: Ex4: Find 3 x 3x 5x f if without dividing. 3 x 3x 5x 1. f x 3x x 5x 6x 1 using synthetic substitution. Using the polynomial in standard form, write the coefficients in a row. Put the x-value to the upper left. Bring down the first coefficient, then multiply by the x-value. Add straight down the columns, and repeat. The number in the bottom right is the value of So : f () f. This also means that 3, 137 is an ordered pair that would be a point on the graph. And the remainder when dividing f x by x 3 is Ex 5: Find f 3 if f x x x x 7 11 using synthetic substitution. This polynomial function is in standard form, however it is missing two terms. We can f x x 0x x 7x 0x 11 to fill in the missing rewrite the function as terms. Page 11 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

12 Precalculus Notes: Unit Polynomial Functions Synthetic Division: Ex.6 a) (x 3 3x 5x 1) (x 3) b) Divide x 4 8x x 6 by x + 3. f x Qx a x... a x a x P n x 1 Recall: n1 1 Possible Rational Zeros of a Polynomial Function = factors of constant factors of leading coefficient P Q This also means that 5 could not be a zero, 7 could not be a zero, 1 could not be a zero,... Ex7: Find the possible rational zeros of f x x x ( ) 7 1 Step 1: The leading coefficient is. is the only factor of. Step : The constant is. All of the factors of are Step 3: List the possible factors - *If we tested for actual zeros using synthetic substitution we would find that 3 and 4 are zeros. Descartes Rule of Signs: The number of real zeros equals the number of sign changes in f x or that number less a multiple of. The number of negative real zeros equals the number of sign changes in f x or that number less a multiple of. Ex8: Use Descartes Rule of Signs to determine the possible number of positive and negative 3 real zeros of the function f x 3x 4x 5x. Number of sign changes of 3 f x : 3 Number of sign changes of f x : f x 3x 4x 5x 1 sign change f x 3x 4x 5x sign changes There is 1 possible positive real zero and or 0 possible negative real zeros. Page 1 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

13 Precalculus Notes: Unit Polynomial Functions Upper and Lower Bound Rules: used to determine if there is no zero larger or smaller than a number c, if the leading coefficient is POSITIVE an 0, using synthetic division for f x x c Upper Bound: c is an upper bound for the real zeros of f if c 0 and every number in the last f x x c using synthetic division has signs that are all nonnegative. line of Lower Bound: c is a lower bound for the real zeros of f if c 0 and every number in the last line f x x c using synthetic division has signs that are alternately nonnegative and f : of nonpositive. Ex9: Verify that all of the real zeros of the function 3 interval 3,. f x 3x 4x 5x lie on the c 0 and signs are all nonnegative, so c is an upper bound of the zeros f 3 : c 3 0 and signs alternate, so c 3 is an lower bound of the zeros Conclusion: Finding the Real Zeros of a Polynomial: f x 3x 4x 5x Ex10: Find all real solutions of the polynomial equation. 3 Note: There is one sign change, so Possible Rational Zeros: Note: Remember, if 1 is a zero, x 1 is a factor Synthetic Substitution (try 1): f is a zero, so we can write f x. Factor the quadratic: Solutions: Reflect: How is the Factor Theorem connected to the Remainder Theorem? Page 13 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

14 Date:.4 Complex Numbers Precalculus Notes: Unit Polynomial Functions Objective: Students will use imaginary unit i to write complex numbers. Add, subtract and multiply complex numbers, use complex conjugates to write the quotient of two complex numbers in standard form, plot complex numbers in the complex plane. I. Number Sets C : Complex: a + bi R : Real Imaginary Q : Rational I: Irrational Z : Integers Transcendental W: Whole N : Natural Transcendental numbers are not the roots of any algebraic equations. All other complex numbers are algebraic. Powers of i: Cont to i 7 i 0 = 1 * 1 1 i i i = -1 i 3 = 4 i Page 14 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

15 Precalculus Notes: Unit Polynomial Functions Mult. complex #s is like FOIL, but i is not a variable. Ex.1 a) (3 + 7i) ( 4i) b) (3 + 7i)( 4i) Ex. a) 3 i3 4i b) ( i) 3 Complex Conjugate of z: (a + bi) is z: (a bi) The are many examples of conjugates, the difference of two squares for example. Ex.3 a) Multiply 3 4i by its conjugate. b) Find x & y. ( + 3i)(x i) = 13 + yi Ex.4 ( i)(4 i) 1i Complex Solutions to Quadratic Equations Discriminant: D < 0 Ex.5 5x + x + 1 = 0 Reflect: Page 15 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

16 Date:.5 The Fundamental Theorem of Algebra Precalculus Notes: Unit Polynomial Functions Syllabus Objectives: 1.7 The student will solve problems using the Fundamental Theorem of Algebra. 1.8 The student will calculate the complex roots of a given function. Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (some zeros may repeat) Complex zeros come in conjugate pairs Review: a bi conjugate a bi A polynomial function with odd degree and real coefficients has at least one real zero Calculator Exercise: State how many complex zeros the function has. Use a graphing calculator for b & c to determine how many of the zeros are real. a) f(x) = x 1x + 19 b) f(x) = 5x 3 + 4x + 3x 4 c) f(x) = x 5 + x 4 x + x 3 18x + 55 Ex1: Find all the zeros of the function and write the polynomial as a product of linear factors. 4 3 f x x 6x 10x 6x 9 Step One: Check for any rational zeros. factors of Possible rational zeros:,, factors of Use synthetic substitution: So is a zero, and is a factor. Step Two: Rewrite the last line as a polynomial and find the zeros of the polynomial. f x 3 x x x Factor by grouping: Step Three: List the zeros. Check that you have the correct number according to the FTA. The polynomial was degree 4, and there are 4 zeros (3 is repeated). Note that i and i are complex conjugates. Page 16 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

17 Precalculus Notes: Unit Polynomial Functions Step Four: Write the polynomial as a product of linear factors. f( x) Ex: Find a polynomial function with real coefficients in standard form that has the zeros and 1 i. Write the polynomial as a product of linear factors using the zeros. Note: If 1 i is a zero, then its complex conjugate, must also be a zero. Write the polynomial in standard form. Irreducible Quadratic Factor: A quadratic factor of a polynomial function is irreducible over the reals if it has real coefficients but no real zeros. Ex3: Write the polynomial as the product of factors that are irreducible over the reals. Then f x 3x x 3x x 60x 0 write the polynomial in completely factored form Step One: Check for any rational zeros. Possible rational zeros: Step Two: Use synthetic substitution: Step Three: Rewrite the last line as a polynomial and find the zeros of the polynomial. Product of factors irreducible over the reals: Step Four: Completely factored form: Page 17 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

18 Precalculus Notes: Unit Polynomial Functions Ex4: Use the given zero to find ALL the zeros of the function. Zero = 1 i ; 4 3 f x x x x 6x 6 Note: By the FTA, we know that f x must have zeros. Since 1 i is a zero, we know that (its conjugate) must also be a zero. Two factors of Multiply: Use long division: f x are: and Zeros of Zeros of f x : You Try: Find a polynomial function with real coefficients in standard form that has the zeros 3, and 1 with multiplicity 3. Reflection: Is the polynomial function found in the You Try unique? Explain your answer. Page 18 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

19 Date:.6 Rational Functions and Asymptotes Precalculus Notes: Unit Polynomial Functions Syllabus Objectives: 1.9 The student will solve rational equations in one variable. Rational Function: the ratio of two polynomial functions r x x f, g x 0 g x Domain of a Rational Function: the set of all real numbers except the zeros if its denominator Recall: Reciprocal Function f x 1 x Rational Functions that are Transformations of the Reciprocal Function 1. gx Ex1: Describe the transformations on f 3 x 1 3 g x x 1 Domain: All reals, x x. gx 4. gx x. State the domain. gx 3 x 1 3 x 1 Using Long Division (used when variable is in the numerator and denominator) Ex: Describe the transformations on f 3 Long division: x 5 3x 4 3x x 5 11 x 5 1 x x. State the domain. gx g x 11 3 x 5 1. gx 3. g x 3. gx 3 4. g x 3 x 1 4 3x x x x 5 Page 19 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

20 Precalculus Notes: Unit Polynomial Functions Recall: Limit: the value that f Notation: lim f lim xa xa lim xa f f x x x x approaches as x approaches a given value The limit as x approaches a of f The limit of f The limit of f x. (From both sides.) x as x approaches a from the left. x as x approaches a from the right. Ex3: Evaluate the limits based on the graph. 1) lim f x x1 ) lim f x x1 3) lim f x x 4) lim f x x End Behavior Asymptotes of f x ax n bx m n m If n m: horizontal asymptote y 0 an If n m: horizontal asymptote y (ratio of the leading coefficients) b n If n m: No horizontal asymptote. End behavior is the function that is the result after performing the division on f x. Note: If the n is one more than m, then the function describing end behavior is linear. This is called a slant asymptote. Vertical Asymptote: Set the denominator equal to zero. Page 0 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

21 Precalculus Notes: Unit Polynomial Functions Ex4: Describe the end behavior of the following functions. a.) b.) y 3 x x x Long Division: 3 1 x 1 degree of the numerator > degree of the denominator y x 4x 5 x 1 6 End Behavior: Approaches the graph of the function 6x y 3 x 1 y x x 4 5 degree of the numerator = degree of the denominator Lead coefficient of numerator = 3 x 1 3 4x 4x 1 1x 1x 3 Lead coefficient of denominator = : End Behavior: horizontal asymptote of y c.) y x x x 3 d.) 6x y 3x Slant: Ex. 5 Find the horizontal and vertical asymptotes: 6x y x 4 Reflection: Page 1 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

22 Date:.7 Graphs of Rational Functions Precalculus Notes: Unit Polynomial Functions Syllabus Objectives.9 The student will sketch the graph of a polynomial, radical, or rational function. Graphing Rational Functions: To sketch the graph of a rational function, find the domain, asymptotes, and intercepts. The plot points in each region created by the asymptotes to locate the curve and use the asymptotes to guide the sketching of the curve. x x Ex1: Graph gx x 1. Domain: Vertical Asymptote: x-intercepts (zeros of numerator): y-intercepts ( g 0 ): g 0 End Behavior: asymptote ; long division results in Slant Asymptote: Plot a few points to graph the lower part of the curve. Note: lim 0 x x 1 g x x x 1 x 8x6 Ex: Graph g x. x 4 Domain: Vertical Asymptotes: x-intercepts (zeros of numerator): y-intercepts ( g 0 ): 0 g End Behavior: horizontal asymptote y Page of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

23 Precalculus Notes: Unit Polynomial Functions Removable Discontinuity: a rational function has a hole in the graph if one or more of the linear factors in the denominator can be cancelled with one of the factors in the numerator. Domain: Ex3: Sketch the graph of g x x 18 x 3x Vertical Asymptotes: Note: x is NOT a vertical asymptote. It is a hole. x-intercepts (zeros of numerator): 0 y-intercepts ( g ): End Behavior: horizontal asymptote y You Try: Sketch the graph of y x x. Describe all relevant features. x 5 QOD: Can a function cross a vertical asymptote? Explain. Can a function cross a horizontal asymptote? Explain. Reflection: Page 3 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

24 Date: Solve Rational Functions Precalculus Notes: Unit Polynomial Functions Syllabus Objective: 1.9 The student will solve rational equations in one variable. Solving a Rational Equation: 1. Multiply every term by the LCD to wipe out the fractions.. Solve the resulting equation. 3. Check for extraneous solutions in the original equation, which can result when multiplying by a variable expression. Ex1: Solve the equation y y y 4 Factor to find LCD: Multiply by LCD: Solve the resulting equation: Check for extraneous solutions: y makes one or more of the denominators in the original equation equal zero, so it is extraneous. Therefore, there is. x 3 Ex: Solve the equation: 5x4 3x8 This equation is a proportion, so we will cross multiply. Solve the equation by factoring: Check for extraneous solutions: neither solution results in a denominator of zero in the original equation Page 4 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

25 Precalculus Notes: Unit Polynomial Functions 3 x 4 Ex3: Find the solution set of the equation x 7x 10 x 5. Factor to find LCD: Multiply by LCD: Solve the resulting equation: Check for extraneous solutions: x does not make one or more of the denominators in the original equation equal zero, so it is a solution. x does make one or more of the denominators in the original equation equal zero, so it is an extraneous solution. Solution: x x 3 4 You Try: Solve the equation x x 3 x x 6. Reflection: Explain two ways you can check your solution graphically when solving a rational equation. Page 5 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

26 Precalculus Notes: Unit Polynomial Functions Sample SAT Question(s): Taken from College Board online practice problems. 1. The projected sales volume of a video game cartridge is given by the function s p 3000 p a, where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 100,000 cartridges are sold at $10 per cartridge, how many cartridges will be sold at $0 per cartridge? (A) 0,000 (B) 50,000 (C) 60,000 (D) 150,000 (E) 00,000. a c 1 S. If 0 a b c d e in the equation given, then the greatest increase in S b d e would result from adding 1 to the value of which variable? (A) a (B) b (C) c (D) d (E) e 3. If is defined for all positive numbers a and b by a b = ab, then 10 = a b (A) 5 3 (B) 5 (C) 5 (D) 0 3 (E) 0 Page 6 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

27 Date: Bonus: Solve Inequalities Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.10 The student will solve and graph inequalities in one variable. Solving an Inequality: 1. Find the zeros of the corresponding equation and the zeros of the denominators of any rational expressions (called critical values).. Make a sign chart (choose a test value) to find the correct interval solutions. Ex1: Solve the inequality x x 4 x Critical Values: Sign Chart: + + Solution: Note: If the inequality was x 5x 4 x 8 0, the solution would be 5,8. Ex: Solve the inequality: Critical Values: 3x x Sign Chart: Solution: Note: x 3 is not in the domain of the original function. Ex3: Solve the inequality. Rewrite the inequality: 1 1 x1 x Critical Values: + + Sign Chart: Solution: Page 7 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

28 Precalculus Notes: Unit Polynomial Functions Solving an Inequality Graphically Ex4: Solve graphically: 3 x 3x 4x 1 0 Find the zeros (watch for hidden zeros!): Note: The nd and 3 rd graphs were created by zooming in to the local minimum 3 times! Solution: Application Problem Ex5: A cylindrical can must have a volume of 58 cubic inches. What dimensions can you use if you must keep the surface area below 100 square inches? Volume: V r h 58 Surface Area: SA r rh 100 Solve for h in the volume equation and substitute into the surface area inequality h r r 100 r 100 r r r Solve by graphing: Solution: Radius must be between and inches. Height must be between 1.87 and inches. You Try: Solve the inequality x x 1 x 0. 4x4 QOD: Explain why you need only choose one test point within the intervals created by the critical value. Reflection: Page 8 of 8 Precalculus Graphical, Numerical, Algebraic: Chapter

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