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Chapter 2 Polynomial, Power, and Rational Functions Group Practice 1. Write an equation in slope-intercept form for a line with slope m 2 and y -intercept 10. 2. Write an equation for the line containing the points ( 2,3) and (3,4). 3. Expand ( x 6) . 4. Expand (2 x 3) . 5. Factor 2 x 8 x 8. 2 2 2 Slide 2- 3 Polynomial Function Let n be a nonnegative integer and let a , a , a ,..., a , a 0 1 n 1 2 n be real numbers with a 0. The function given by f ( x) a x a x ... a x a x a n 1 n n n is a polynomial function of degree n. The leading coefficient is a . n n 1 2 2 1 0 Note: Remember Polynomial Poly = many Nomial= terms So it literally means “many terms” Name Form Zero Function Undefined Degree f(x)=0 Constant Function f(x)=a (a≠0) 0 Linear Function 1 f(x)=ax+b (a≠0) Quadratic Function f(x)=ax2+bx+c (a≠0) 2 Remember The highest power (or highest degree) tells you what kind of a function it is. Example #1 Which of the follow is a function? If so, what kind of a function is it? A) B) 𝑓 𝑥 = 3𝑥 2 − 5𝑥 + 6 7 𝑓 𝑥 = 𝑥 −4 + 3𝑥 + 4 C) p x = 9𝑥 4 + 25𝑥 2 D) 𝑦 = 15𝑥 − 2𝑥 6 E) 𝑡 𝑥 = 8𝑥 6 Group talk: Tell me everything about linear functions Slide 2- 9 Average Rate of Change (slope) The average rate of change of a function y f ( x) between x a and x b, a b, is f (b) f (a) . ba Rate of change is used in calculus. It can be expressing miles per hour, dollars per year, or even rise over run. Write an equation for the linear function f such that f (-1) 2 and f (2) 3. Example Answer Use point-slope form (-1,2) (2,3) Y-3=(1/3)(x-2) Ultimate problem In Mr. Liu’s dream, he purchased a 2014 Nissan GT-R Track Edition for $120,000. The car depreciates on average of $8,000 a year. 1)What is the rate of change? 2)Write an equation to represent this situation 3) In how many years will the car be worth nothing? Answer 1) -8000 2) y=price of car, x=years y= −8000𝑥 + 120000 3) When the car is worth nothing y=0 X=15, so in 15 years, the car will be worth nothing. Ultimate problem do it in your group (based on 2011 study) When you graduate from high school, the starting median pay is $33,176. If you pursue a professional degree (usually you have to be in school for 12 years after high school), your starting median pay is $86,580. 1) Write an equation of a line relating median income to years in school. 2) If you decide to pursue a bachelor’s degree (4 years after high school), what is your potential starting median income? Answer 1) y=median income, x=years in school Equation: y= 4450.33x+33176 2) Since x=4, y=50,977.32 My potential median income is $50,977.32 after 4 years of school. You are saying more school means more money?!?! Slide 2- 17 Characterizing the Nature of a Linear Function Point of View Characterization Verbal degree 1 polynomial of Algebraic (m≠0) f(x) = mx + b Graphical slope m and y-intercept b slant line with Analytical constant function with Linear Correlation When you have a scatter plot, you can see what kind of a relationship the dots have. Linear correlation is when points of a scatter plot are clustered along a line. Slide 2- 19 Linear Correlation Slide 2- 20 Properties of the Correlation Coefficient, r 1. 2. 3. 4. 5. -1 ≤ r ≤ 1 When r > 0, there is a positive linear correlation. When r < 0, there is a negative linear correlation. When |r| ≈ 1, there is a strong linear correlation. When |r| ≈ 0, there is weak or no linear correlation. Slide 2- 21 1. 2. 3. 4. Enter and plot the data (scatter plot). Find the regression model that fits the problem situation. Superimpose the graph of the regression model on the scatter plot, and observe the fit. Use the regression model to make the predictions called for in the problem. Regression Analysis Group Work: plot this with a calculator. Example of Regression Price per Box Boxes sold 2.40 38320 2.60 33710 2.80 28280 3.00 26550 3.20 25530 3.40 22170 3.60 18260 Slide 2- 23 Group Work Describe how to transform the graph of f ( x) x into the graph of 2 f ( x) 2 x 2 3. 2 Answer Horizontal shift right 2 Vertical shift up 3 Vertical stretch by a factor of 2 or horizontal shrink by a factor of 1/2 Group Work Describe 𝑓 𝑥 =− the transformation 3 2 𝑥+4 2 +6 Answer Horizontal shift left 4 Vertical shift up 6 Vertical stretch by a factor of 3/2 or horizontal shrink by a factor of 2/3 reflect over the x-axis Slide 2- 27 Vertex Form of a Quadratic Equation Any quadratic function f(x) = ax2 + bx + c, a≠0, can be written in the vertex form f(x) = a(x – h)2 + k The graph of f is a parabola with vertex (h,k) and axis x = h, where h = -b/(2a) and k = c – ah2. If a>0, the parabola opens upward, and if a<0, it opens downward. Group Work: where is vertex? 3 2 𝑓 𝑥 =− 𝑥+4 𝑓 𝑥 =2 𝑥−1 2 2 +6 −3 Answer (-4,6) (1,-3) Example: Use completing the square to make it into vertex Use the vertex form of a quadratic function to find the vertex and axis form of the graph of f ( x) 2 x 8 x 11. Rewrite the equation in vertex form. 2 Group Work Change 𝑓 this quadratic to vertex form 𝑥 = −3𝑥 2 + 6𝑥 − 5 Answer 𝑓 𝑥 = −3 𝑥 − 1 2 −2 Slide 2- 33 Characterizing the Nature of a Point of View Characterization Quadratic Function Verbal polynomial of degree 2 Algebraic f ( x) ax bx c or f ( x) a ( x - h) k (a 0) 2 2 Graphical parabola with vertex (h, k ) and axis x h; opens upward if a > 0, opens downward if a < 0; initial value = y -intercept = f (0) c b b 4ac x-intercepts 2a 2 Slide 2- 34 Vertical Free-Fall Motion The height s and vertical velocity v of an object in free fall are given by 1 s (t ) gt v t s and v(t ) gt v , 2 where t is time (in seconds), g 32 ft/sec 9.8 m/sec is the acceleration 2 0 0 0 2 2 due to gravity , v is the initial vertical velocity of the object, and s is its 0 initial height. 0 Example You are in MESA and we are doing bottle rockets. You launched your rocket and its’ total time is 8.95 seconds. Find out how high your rocket went (in meters) Flyin’ High Answer You first have to figure out how fast your rocket is when launched. Remember the velocity at the max is 0. Also the time to rise to the peak is one-half the total time. So 8.96/2 = 4.48s 0 = −9.8 4.48 + 𝑣0 𝑣0 = 43.904𝑚/𝑠 𝑠 4.48 = − 𝑠 4.48 = 98.34 𝑚 1 2 9.8 4.48 2 + 43.904 4.48 + 0 Homework Practice Pg 182-184 #1-12, 45-50 Pgs 182-184 #14-44e, 55, 58,61 Power Functions with Modeling Slide 2- 39 Power Function Any function that can be written in the form f(x) = k·xa, where k and a are nonzero constants, is a power function. The constant a is the power, and the k is the constant of variation, or constant of proportion. We say f(x) varies as the ath power of x, or f(x) is proportional to the ath power of x. Group Work Write the following expressions using only positive integer powers. 1. x 2. r 3. m 5/3 -3 1.5 Write the following expressions in the form k x using a single rational number for the power of a. a 4. 16 x 5. 3 x 27 3 Group Work: Answer the following with these two functions 3 𝑓 𝑥 = Power: Constant of variation: Domain: Range: Continuous: Increase/decrease: Symmetric: Boundedness: Max/min: Asymptotes: End behavior: 𝑥 𝑓 𝑥 = 𝑥 −2 State the power and constant of variation for the function f ( x) x , and graph it. 4 Example Analyzing Power Functions State the power and constant of variation for the function f ( x) x , and graph it. 4 f ( x) x x 1 x so the power is 1/4 and the constant of variation is 1. 4 1/ 4 1/ 4 Slide 2- 43 Slide 2- 44 Monomial Function Any function that can be written as f(x) = k or f(x) = k·xn, where k is a constant and n is a positive integer, is a monomial function. Slide 2- 45 Example Graphing Monomial Functions Describe how to obtain the graph of the function f ( x) 3x from the graph of g ( x) x with the same power n. 3 n Example Graphing Monomial Functions Describe how to obtain the graph of the function f ( x) 3x from the graph of g ( x) x with the same power n. 3 n We obtain the graph of f ( x) 3x by vertically stretching the graph of g ( x) x by a factor of 3. Both are odd functions. 3 3 Slide 2- 46 Note: Remember, it is important to know the parent functions. Everything else is just a transformation from it. Parent functions can be found in chapter 1 notes. Group Talk: What are the characteristics of “even functions”? What are the characteristics of “odd functions”? What happen to the graphs when denominator is undefined? Clue: Look at all the parent functions. Slide 2- 49 For any power function f(x) = k·xa, one of the following three things happens when x < 0. f is undefined for x < 0. f is an even function. f is an odd function. Graphs of Power Functions Slide 2- 50 Graphs of Power Functions Determine whether f is even, odd, or undefined for x<0 Usually, it is easy to determine even or odd by looking at the power. It is a little different when the power is a fraction or decimal. When the power is a fraction or decimal, you have to determine what happened to the graph when x<0 Example: Determine whether f is even, odd, or undefined for x<0 1)𝑓 𝑥 = 𝑥 1 4 2)𝑓 𝑥 = 𝑥 2 3 3)𝑓 𝑥 = 𝑥 4 3 Answer 1) undefined for x<0 2) even 3) even There is a trick! Odd functions can only be an integer! It can not be a fraction except when denominator is 1 Even functions is when the numerator is raised to an even power. Can be a fraction If the power is a fraction and the numerator is an odd number, it is undefined x<0 Homework Practice Pgs 196-198 #1-11odd, 17, 27, 30, 31, 39, 43, 48, 55, 57 Polynomial Functions of Higher degree with modeling Group Talk How do you determine how many potential solutions you have on a graph? What is a polynomial? Polynomial means “many terms” Each monomial in the sum a x , a x ,..., a is a term of the polynomial. n 1 n n n 1 0 A polynomial function written in this way, with terms in descending degree, is written in standard form. The constants a , a ,..., a are the coefficients of the polynomial. n n 1 0 The term a x is the leading term, and a is the constant term. n n 0 Group Review: Shifts 𝑓 𝑥 = −6 −𝑥 + 8 What 2 −5 is the parent function? Give me all the shifts! Answer Parent function is 𝑓 𝑥 = 𝑥 2 Note: You have to factor out the negative in front of 2 the x 𝑓 𝑥 = −6 − 𝑥 − 8 −5 Horizontal shift right 8 Vertical shift down 5 Vertical Stretch by factor of 6 Horizontal shrink by a factor of 1/6 Flip over the x axis Flip over the y axis Group Work Describe how to transform the graph of an appropriate monomial function f ( x) a x into the graph of h( x) ( x 2) 5. Sketch h( x) and n n compute the y -intercept. 4 Example Graphing Transformations of Monomial Functions Describe how to transform the graph of an appropriate monomial function f ( x) a x into the graph of h( x) ( x 2) 5. Sketch h( x) and n 4 n compute the y -intercept. You can obtain the graph of h( x ) ( x 2) 5 by shifting the graph of f ( x) x two units to the left and five units up. The y -intercept of h( x) 4 4 is h(0) 2 5 11. 4 Slide 2- 62 Remember End Behavior? What is End Behavior? You have to find out the behavior when 𝑥 → ∞ 𝑎𝑛𝑑 𝑥 → −∞ lim 𝑓(𝑥) = 𝑥→∞ lim 𝑓 𝑥 = 𝑥→−∞ Find the end behavior for all! 𝑓 𝑡 = 2𝑡 2 𝑓 𝑎 = −0.5𝑎7 𝑓 𝑔 = 5𝑔−5 𝑓 𝑠 = −3𝑥 15 Think about this one Answer lim 2𝑡 2 = ∞ 𝑡→∞ lim 2𝑡 2 = ∞ 𝑡→−∞ lim −0.5𝑎7 = −∞ 𝑎→∞ lim −0.5𝑎7 = ∞ 𝑎→−∞ lim 5𝑔−5 = 0 𝑔→∞ lim 5𝑔−5 = 0 𝑔→−∞ lim −3𝑥 15 = −3𝑥 15 𝑠→∞ lim −3𝑥 15 = −3𝑥 15 𝑠→−∞ Why do you think this happens? Mr. Liu the trickster Find the end behavior for: 𝑥 = 50𝑥 6 − 180𝑥 5 + 98𝑥 4 − 140𝑥 3 + 10𝑥 2 − 𝑥 + 35 𝑓 Answer You only look at the term with the highest power, which is the 6th power lim 𝑓(𝑥) = ∞ 𝑥→∞ lim 𝑓 𝑥 = ∞ 𝑥→−∞ Determining if you have a min/max Graph this function 𝑓 𝑡 = 𝑡3 + 𝑡 Tell me about this function Answer It is increasing for all domains Therefore there is no min/max There is one zero at t=0 Determine if you have a min/max Graph this function 𝑓 𝑡 = 𝑡3 − 𝑡 Tell me about this function Answer Graph increases from (−∞, −0.38) Graph decreases from −0.38,0.58 Graph increases from (0.58, ∞) Therefore there is a local max at x=-0.38 There is a local min at x=0.58 Three zeros: x=-1, x=0 and x=1 Slide 2- 73 Potential Cubic Functions (what it can look like) Slide 2- 74 Quartic Function (what it can look like) Slide 2- 75 Local Extrema and Zeros of A polynomial function of degree n has at most Polynomial Functions n – 1 local extrema and at most n zeros. For example If you have a function that is to the 3rd power You may have potential of 3 zeros (3 solutions) You may have 2 local extrema (either max or min) Now try this! Function Zeros? Extremas? Function to the 5th power, how many… to the 4th power, how many… Zeros? Extremas? Remember I asked you guys about the even powers vs odd powers? Here it is! More examples Finding zeros Note: very very very important to know how to factor!!!! Example Solve: 𝑓 𝑥 = 𝑥 3 − 𝑥 2 − 6𝑥 = 0 Group Work Find the zeros of f ( x) 2 x 4 x 6 x. 3 2 Slide 2- 83 Multiplicity of a Zero of a Function If f Polynomial is a polynomial function and x c is a factor of f m but x c m 1 is not, then c is a zero of multiplicity m of f . Slide 2- 84 Example Sketching the Graph of a Factored Polynomial Sketch the graph of f ( x) ( x 2) ( x 1) . 3 2 Slide 2- 85 Intermediate Value If aTheorem and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b]. Note: That is important for Calculus! Homework Practice Pgs 209-210 #3, 6, 15-36, multiple of 3 Real zeros of polynomial Functions What’s division? There are two ways to divide polynomials Long division Synthetic division Example: 2𝑥 4 − 𝑥 3 − 2 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 2𝑥 2 + 𝑥 + 1 Work: Group Work Use long division to find the quotient and remainder when 2 x x 3 is divided by x x 1. 2 4 3 2x x 1 x x 1 2x x 0x 0x 3 2x 2x 2x x 2x 0x 3 2 Answer 2 4 3 2 4 3 2 3 2 x x x 3 x 2x 4 2 2 2 x + x3 x 1 2x 2 x 3 x x 1 2 x x 1 3 2 2 2x 2 x x 1 2 Remainder theorem If polynomial f ( x) is divided by x k , then the remainder is r f (k ). What does the remainder theorem say? Well, it tells us what the remainder is without us doing the long division! Basically, you substitute what make the denominator 0! EX: if it was x-3, then you substitute x=3, so it’s f(3)=r I am so happy such that I don’t have to do the long division to find the remainder! Example: Find the remainder when f ( x) 2 x x 12 is divided by x 3. 2 Answer r f (3) 2 3 3 12 =33 2 Group Work Find the 6𝑥 3 − 5𝑥 remainder + 5 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 𝑥 − 4 Synthetic Division Divide 2𝑥 3 − 3𝑥 2 − 5𝑥 − 12 𝑏𝑦 𝑥 − 3 Divide 3x 2 x Work x 5 by x 1 using synthetic division. Group 3 2 Slide 2- 103 Example Using Synthetic Divide 3x 2 x x 5 by x 1 using synthetic division. Division 3 1 3 2 2 1 5 1 3 3 3 3x 2 x x 5 3 3x x 2 x 1 x 1 3 2 2 2 3 1 1 1 2 5 2 3 Again Divide 𝑥 8 − 1 𝑏𝑦 𝑥 + 2 Rational Zeros Theorem This is P/Q Slide 2- 106 Rational Zeros Theorem Suppose f is a polynomial function of degree n 1 of the form f ( x) a x a x ... a , with every coefficient an integer n 1 n n n 1 0 and a 0. If x p / q is a rational zero of f , where p and q have 0 no common integer factors other than 1, then p is an integer factor of the constant coefficient a , and 0 q is an integer factor of the leading coefficient a . n In Another word P are the factors of the last term of the polynomial Q are the factors of the first term of the polynomial Use Synthetic division to determine if that is a zero Example: 𝑓 𝑥 = 𝑥 3 − 3𝑥 2 + 1 Group Work: Find Rational Zeros 𝑓 𝑥 = 3𝑥 3 + 4𝑥 2 − 5𝑥 − 2 Slide 2- 110 Example Finding the Real Zeros of a Polynomial Function Find all of the real zeros of f ( x) 2 x 7 x 8 x 14 x 8. 4 3 2 Finding the polynomial Degree 3, with -2,1 and 3 as zeros with coefficient 2 Answer 2(x+2)(x-1)(x-3) Group Work Find polynomial with degree 4, coefficient of 4 with 0, ½, 3 and -2 as zeros Answer 4x(x-1/2)(x-3)(x+2) Homework Practice Pgs 223-224 # 1, 4, 5, 7, 15, 18, 28, 35, 36, 49, 50, 57 Complex Zeros and the Fundamental Theorem of Algebra Bell Work Perform the indicated operation, and write the result in the form a bi. 1. 2 3i 1 5i 2. 3 2i 3 4i Factor the quadratic equation. 3. 2 x 9 x 5 Solve the quadratic equation. 4. x 6 x 10 0 List all potential rational zeros. 5. 4 x 3 x x 2 2 2 4 2 Slide 2- 118 Fundamental Theorem of A Algebra polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated. Slide 2- 119 Linear Factorization If f (Theorem x) is a polynomial function of degree n 0, then f ( x) has precisely n linear factors and f ( x) a( x z )( x z )...( x z ) where a is the 1 2 n leading coefficient of f ( x) and z , z ,..., z are the complex zeros of f ( x). 1 2 n The z are not necessarily distinct numbers; some may be repeated. i In another word The highest degree tells you how many zeros you should have (real and nonreal) and how many times it may cross the xaxis (solutions) Very Important!!! If you have a nonreal solution, it comes in pairs. One is the positive and one is negative (next slide is an example) Example: Find the polynomial 𝑓 𝑥 = (𝑥 − 4𝑖)(𝑥 + 4𝑖) Note: This is linear factorization How many real zeros? How many nonreal zeros? What’s the degree of polynomial? Group Work: Find the polynomial 𝑓 𝑡 = (𝑡 − 1)(𝑡 + 2)(𝑡 − 𝑖)(𝑡 + 𝑖) Note: This is called linear factorization How many real zeros? How many nonreal zeros? What’s the degree of polynomial? Group work Find the polynomial with -1, 1+i, 2-i as zeros Answer (x+1)(x-(1+i))(x+(1+i))(x-(2-i))(x+(2-i)) (x+1)(x-1-i)(x+1+i)(x-2+i)(x+2-i) or Slide 2- 125 Group Work Write a polynomial of minimum degree in standard form with real coefficients whose zeros include 2, 3, and 1 i. Group work: Finding Complex Zeros Z=1-2i is a zero of 𝑓 𝑥 4𝑥 4 + 17𝑥 2 + 14𝑥 + 65 Find the remaining zeros and write it in its linear factorization Write the function as a product of linear factorization and as real coefficient 𝑓 𝑥 = 𝑥 4 + 3𝑥 3 − 3𝑥 2 + 3𝑥 − 4 Answer (x-1)(x+4)(x-i)(x+i) As Real coefficient (𝑥 − 1)(𝑥 + 4)(𝑥 2 + 1) Slide 2- 129 Example Factoring a Polynomial Write f ( x) 3x x 24 x 8 x 27 x 9 as a product of linear and irreducible quadratic factors, each with real coefficients. 5 4 3 2 Slide 2- 130 Example Factoring a Write f ( x) 3x x 24 x 8 x 27 x 9 as a product of linear and Polynomial irreducible quadratic factors, each with real coefficients. 5 4 3 2 The Rational Zeros Theorem provides the candidates for the rational zeros of f . The graph of f suggests which candidates to try first. Using synthetic division, find that x 1/ 3 is a zero. Thus, f ( x) 3 x x 24 x 8 x 27 x 9 5 4 3 2 1 x 3 x 8 x 9 3 1 3 x x 9 x 1 3 4 2 2 2 1 3 x x 3 x 3 x 1 3 2 Homework Practice Pg 234 #1, 3, 5, 14, 17-20 ,37, 38, 6, 11, 15, 21, 23, 27-29, 33,43, 51 Graphs of Rational Functions Slide 2- 133 Rational Functions Let f and g be polynomial functions with g ( x) 0. Then the function f ( x) given by r ( x) is a rational function. g ( x) Note: Vertical Asymptote You look at the restrictions at the denominator to determine the vertical asymptote Slide 2- 135 Group Work Find the domain of f and use limits to describe the behavior at value(s) of x not in its domain. 2 f ( x) x2 Answer Remember you always see what can’t X be (look at the denominator) D: −∞, −2 𝑢(−2, ∞) Note: Horizontal Asymptote If the power of the numerator is < power of denominator then horizontal asymptote is y=0 If the power of the numerator is = power of denominator then horizontal asymptote is the coefficient If the power of numerator is > power of denominator, then there is no horizontal asymptote Note 2 If numerator degree > denominator degree. You may have a slant asymptote. You have to use long division to determine the function Example: Find the horizontal asymptote 𝑓 𝑓 𝑓 𝑥 = 3𝑥 2 −5 5𝑥 3 𝑥 = 𝑥7 𝑥5 𝑥 = 6𝑥 3 +2𝑥+5 𝑥 3 −5𝑥+6 Answer Y=0 None Y=6 Slant asymptote example 𝑓 𝑥 = 𝑥3 𝑥 2 −9 Slide 2- 142 Example Finding Asymptotes of Rational Functions Find the asymoptotes of the function f ( x) 2( x 3)( x 3) . ( x 1)( x 5) Slide 2- 143 Example Finding Asymptotes 2( x 3)( x 3) Find the of the function f ( x) . ofasymoptotes Rational Functions ( x 1)( x 5) There are vertical asymptotes at the zeros of the denominator: x 1 and x 5. The end behavior asymptote is at y 2. Example Graphing a Rational Function x 1 Find the asymptotes and intercepts of f ( x) x 2 x 3 and graph f ( x). Slide 2- 144 Example Graphing a Rational Function x 1 Find the asymptotes and intercepts of f ( x) x 2 x 3 and graph f ( x). The numerator is zero when x 1 so the x-intercept is 1. Because f (0) 1/ 6, the y -intercept is 1/6. The denominator is zero when x 2 and x 3, so there are vertical asymptotes at x 2 and x 3. The degree of the numerator is less than the degree of the denominator so there is a horizontal asymptote at y 0. Slide 2- 145 Ultimate Problem 3𝑥 2 −2𝑥+4 𝑥 2 −4𝑥+5 𝑓 𝑥 = Domain: Range: Continuous: Increase/decrease: Symmetric: Y-intercept: X-intercept: Boundedness: Max/min: Asymptotes: End behavior: Homework Practice Pg 245 #3, 7, 11-19, 21, 23, 25 Solving Equations and inequalities Slide 2- 149 Example Solving by Clearing 2 Solve x 3. x Fractions Slide 2- 150 Example Eliminating 1 2x 2 Solve the equation . x 3 x 1Solutions x 4x 3 Extraneous 2 Group Work 𝑥 4 𝑥 + = 10 Group Work 2𝑥 1 + 𝑥−1 𝑥−3 = 2 𝑥 2 −4𝑥+3 Slide 2- 153 Example Finding a Minimum Find the dimensions of the rectangle with minimum perimeter if its area is 300 squarePerimeter meters. Find this least perimeter. Solving inequalities Solving inequalities, it would be good to use the number line and plot all the zeros, then check the signs. Slide 2- 155 Example Finding where a Polynomial is Zero, Positive, or Let f ( x) ( x 3)( x 4) . Determine the real number values of x that Negative cause f ( x) to be (a) zero, (b) positive, (c) negative. 2 Slide 2- 156 Example Solving a Polynomial Solve x 6 x 2 8 x graphically. Inequality Graphically 3 2 Slide 2- 157 Example Solving a Polynomial Solve x 6 x 2 8 x graphically. Inequality Graphically 3 2 Rewrite the inequality x 6 x 8 x 2 0. Let f ( x) x 6 x 8 x 2 and find the real zeros of f graphically. 3 2 3 2 The three real zeros are approximately 0.32, 1.46, and 4.21. The solution consists of the x values for which the graph is on or below the x-axis. The solution is (, 0.32] [1.46, 4.21]. Slide 2- 158 Example Creating a Sign Chart x 1 Let r ( x) . Determine the values of x that cause r ( x) to be 3Rational x 1 for x a Function (a) zero, (b) undefined, (c) positive, and (d) negative. Slide 2- 159 Example Solving an Inequality Solve ( x 2) x 1 0. Involving a Radical Group Work 𝑓 𝑥 = 2𝑥+1 (𝑥+3)(𝑥−1) determine when it’s a) zero b) undefined c) positive d) negative Group Work Solve 5 𝑥+3 − 2 𝑥−2 >0 Group Work 𝑠𝑜𝑙𝑣𝑒 𝑥−8 𝑥−2 ≤0 Group Work (𝑥 + 2) 𝑥 ≥ 0 Homework Practice 253-254 35, 39 264 #3, 9, 11,15, 17, 27, 28, 31, 32, 34, #1, 6, 8, 13, 21, 28, 33, 36, 47