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- 1. Functions and Their Graphs<br />Continuation<br />
- 2. Types of Functions<br />
- 3. 5. Quadratic Functions<br />A quadratic function is a function of the form <br />f(x) = ax2 +bx +c where a, b and c are real numbers and a ≠ 0.<br />Domain: the set of real numbers<br />Graph: parabola<br />Examples: parabolas parabolas<br /> opening upward opening downward<br />
- 4. Graphs of Quadratic Functions<br />The graph of any quadratic function is called a parabola. Parabolas are shaped like cups, as shown in the graph below. <br />If the coefficient of x2 is positive, the parabola opens upward; otherwise, the parabola opens downward. <br />The vertex (or turning point) is the minimum or maximum point. <br />
- 5. Graphing Parabolas <br />Given f(x) = ax2+ bx+c<br />Determine whether the parabola opens upward or downward. If a >0, it opens upward. If a < 0, it opens downward.<br />Determine the vertex of the parabola. The vertex is <br />The axis of symmetry is <br />The axis of symmetry divides the parabola into two equal parts such that one part is a mirror image of the other.<br />
- 6. Graphing Parabolas <br />Given f(x) = ax2+ bx+c<br />Find any x-intercepts by replacing f (x) with 0. Solve the resulting quadratic equation for x. <br />Find the y-intercept by replacing x with zero. <br />Plot the intercepts and vertex. Connect these points with a smooth curve that is shaped like a cup.<br />
- 7. Example 9.<br />The function f(x) = 1 - 4x - x2 has its vertex at _____.<br /> <br />A. (2,11) <br />B. (2,-11) <br />C.( -2,-3) <br />D.(-2,1)<br />
- 8. Example 10a.<br />Identify the graph of the given function: y = 3x2 - 3. <br />
- 9. Example 10b.<br />Identify the graph of the given function: 4y = x2.<br />
- 10. Example 10c.<br />Identify the graph of the given function: y = (x - 2)(x – 2).<br />
- 11. Minimum and Maximum: Quadratic Functions<br />Consider f(x) = ax2 + bx +c.<br />If a > 0, then f has a minimum that occurs at x = -b/(2a). This minimum value is f(-b/(2a)).<br />If a < 0, the f has a maximum that occurs at x = -b/(2a). This maximum value is f(-b/(2a)).<br />
- 12. Example 11a.<br />The maximum value of the function f(x) = -3x2 – 2x + 4 is ____.<br /> <br />A. 13/3 <br />B. 3/13 <br />C. 9 <br />D. 13<br />
- 13. Example 11b.<br />The function f(x) = x2 – 8x + 16 has _____.<br /> <br />A. a minimum value at x = -4 <br />B. a maximum value at x = -4<br />C. a minimum value at x = 4 <br />D. a maximum value at x = 4<br />
- 14. Example 12.<br />The revenue of a charter bus company depends on the number of unsold seats. If the revenue R(x) is given by R(x) = 5000 + 50x – x2, where x is the number of unsold seats, find the maximum revenue and the number of unsold seats that corresponds to the maximum revenue.<br />Solution:<br /> The revenue function, R(x) = 5000 + 50x – x2 is a quadratic function with a = -1, b = 50 and c = 5000. Since a = -1 < 0, the R(x) has a maximum that occurs at x = -b/(2a) and the maximum value is R(-b/(2a)). <br />
- 15. 6. Absolute Value Functions<br />An absolute value function f is defined by <br />Domain: the set of real numbers<br />Graph: v-shaped<br />Examples: y = -|x| y = |x| y = x - |x|<br />
- 16. 7. Rational Functions<br />A Rational Function is a function in the form:<br />where p(x) and q(x) are polynomial functions and q(x) ≠ 0.<br />Examples: <br />
- 17. 8. Polynomial Functions<br />A polynomial function is a function of the form:<br />n must be a positive integer<br />an ≠ 0<br />All of these coefficients are real numbers<br />The degree of the polynomial is the largest power on any x term in the polynomial.<br />
- 18. Examples:Graphs of Polynomial Functions<br />
- 19. Polynomial Functions and Equations<br />
- 20. The Remainder Theorem<br />If P(x) is a polynomial and r is a real number, then if P(x) is divided by x – r, the remainder is P( r ).<br />
- 21. Example 1.<br />Find the remainder when (2x3 – 3x2 - 4x - 17) is divided by (x – 3). <br />Solution:<br />Applying the Remainder Theorem, we have<br /> P(3) = 2(3)3 – 3(3)2 – 4(3) – 17 <br />= 54 – 27 – 12 – 17 = -2 <br />The remainder is -2.<br />
- 22. Example 2.<br />Use the Remainder Theorem to find the remainder obtained by dividing the polynomial by the given binomial that follows it.<br />a. ; <br />b. ; <br />c. ; <br />Answers: a. -1.5 b. 2 c. 2<br />
- 23. A consequence of the remainder theorem is the Factor Theorem. <br />It enables us to determine whether a specific expression of the form (x – r) is a factor of a given polynomial. <br />
- 24. The Factor Theorem<br />If P(x) is a polynomial and r is a real number, then P(x) has (x – r) as a factor if and only if P( r ) = Q.<br />
- 25. Example 3.<br />Show that (x – 4) is a factor of <br /> (2x3 – 6x2 – 5x – 12).<br />Solution: Applying the Factor Theorem, we have<br />P(x) = 2x3 – 6x2 – 5x – 12<br />P(4) = 2(4)3 – 6(4)2 – 5(4) – 12<br /> = 2(64) – 6(16) – 20 – 12 <br /> = 128 – 96 – 20 – 12 = 0<br />Therefore, by the factor theorem, (x – 4) is a factor of (2x3 – 6x2 – 5x – 12).<br />
- 26. Example 4.<br />Use the Factor Theorem to show that the given binomial is a factor of the polynomial.<br />a. ; <br />b. ; <br />
- 27. Converse of the Factor Theorem<br />If (x – r) is a factor of f(x) then <br /> f(r ) = R = 0, then r is a zero of f(x).<br />
- 28. Example 5.<br />Determine whether (x + 1) is a factor of 5x4 + x3 – 4x2 – 6x – 10.<br />Solution:<br />P(x) = 5x4 + x3 – 4x2 – 6x – 10<br />P(-1) = 5(-1)4 + (-1)3 – 4(-1)2 – 6(-1) – 10<br />= 5 – 1 – 4 + 6 – 10 <br /> = - 4 <br />Since P( -1) 0, then (x + 1) is not a factor of P(x) = 5x4+ x3 – 4x2 – 6x – 10.<br />
- 29. Exercises:<br />Determine whether the linear expression is a factor of P(x):<br />a. x – 2 ; P(x) = 4x3 – 7x2 + x – 2 <br />b. x + 3 ; P(x) = 2x4 + 5x3 + 11x + 6<br />c. x + 2 ; P(x) = x4 + 2x3 – 12x2 – 11x + 6<br />
- 30. References:<br />http://rechneronline.de/function-graphs/<br />http://www.coolmath.com/graphit/<br />

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