+ bx + c = 0, you can solve for x by using The Quadratic Formula. x

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1 Math 33B Intermediate Algebra Fall 01 Name Study Guide for Exam 4 The exam will be on Friday, November 9 th. You are allowed to use one 3" by 5" index card on the exam as well as a scientific calculator. For the exam you need to know how to do the following: 1. Solve quadratic equations by using the square root property. Solve quadratic equations by completing the square. 8.1 * The Square Root Property: If x a, then x a * To solve ax + bx + c = 0 by completing the square: 1) set up the equation so that the variable terms are on the left of the equal sign, in standard form, and the constant term is on the right. Basically, get it into the form ax bx c. ) divide by a, so the coefficient of x is 1. 3) complete the square by taking one-half the coefficient of the x-term, squaring it, and adding this quantity to both sides of the equation. Basically, add b to both sides. 4) factor the Perfect Square Trinomial on the left side of the equation and simplify the right side. Remember, it always factors into x b 5) use the principle of square roots 6) solve the remaining equation. Solve quadratic equations by using the quadratic formula. 8. * For ax b b 4ac + bx + c = 0, you can solve for x by using The Quadratic Formula. x a 3. Solve application problems using various techniques for solving quadratic equations. 8.3 * There are 4 ways that we know how to solve a quadratic equation. 1.) Factoring,.) Square Root Property, 3.) Completing the Square, 4.) Using the Quadratic formula You may use any of these 4 methods to solve a quadratic equation obtained from an application problem. i.e.: finding the time it takes, t, for a projectile to be a certain height, h (t). 4. Solve equations that can be made into quadratic equations using u-substitutions. 8.4 * With the equation written in descending order, let u the middle variable or binomial, then u the first variable or binomial. Rewrite the equation using u to solve, then replace u with what u=, so you can solve what you were originally asked to solve. 5. Graph quadratic equations using the axis of symmetry, the vertex, and the x-intercepts. 8.5 * The axis of symmetry is the vertical line that goes down the middle of the parabola, through the vertex. Since the axis of b symmetry is a line the equation of this line is x a. * The vertex is the minimum point for, or the maximum point for. Since the vertex is a point, its b b coordinates are vertex:, f ( ). a a * The vertex is used to solve application problems that require you to solve for the minimum or maximum values, i.e.: highest height of a projectile, maximum area of a rectangle, minimum cost, etc...

2 * The x-intercepts are the points where the graph crosses the x-axis (where y=0). Each time you set an equation equal to zero and solve for x you are finding the x-intercepts. To determine the number of x- intercepts a quadratic equation has, calculate the discriminant. Discriminant Solutions to ax bx c 0 Graph of b 4ac f ( x) ax bx c The equation has two unequal REAL solutions. Two x-intercepts. If b 4ac 0 If b 4ac is a perfect square, the solutions are RATIONAL numbers.) If b 4ac 0 If b 4ac is NOT a perfect square, the solutions are IRRATIONAL CONJUGATES. The equation has only one REAL solution. It would be a double root and if a, b, and c are rational, it would be a RATIONAL number. One x-intercept. If b 4ac 0 The equation has NO REAL solutions. It has two IMAGINARY solutions. They would be COMPLEX CONJUGATES. No x-intercepts. * Using translations (shifting the graph) is an easy way graph a quadratic equation by putting it into the form f ( x) a( x h) k, with vertex: ( h, k), and a determines whether the graph will be narrower or wider than the original graph. *The greater the absolute value of a, the narrower the graph* 6. Solve quadratic, polynomial, and rational inequalities in one variable. 8.6 * Find the zeros of the inequality and the values that make the function undefined and set them as boundary regions on a number line. Test each region. If the statement is true, shade that region. Write the shaded region in interval notation. Practice Problems The answer to all the problems listed below, even and odd, are in the back of the book. For those of you who have the Chapter Test Prep Video cd that came with the book, you can use it to see someone solving each of the problems in the Chapter Tests. If you don't have it, it is available at the math lab.

3 Math 33B Practice Exam 4: Chapter 8 For problems 1 8, solve for x. Express irrational answers in simplified radical form: 1. 3(x 4) 54 = 0. 3(x + 7) + 36 = 0 3. x 7x 1 = 0 4. x = 3 4x Solve using the appropriate u-substitution: 5. (x + x) 14(x + x) = x 3 + x = 0 7. x 7 x x 3x 10 0 Write a quadratic equation in standard form with the given solution set: 9. 1, { - 8i, 8i} 11. i 3, i 3 1. A company s profit (in thousands of dollars) can be approximated over the next 15 years by the function p ( y) 1.6y 5y 31, where p(y) is the profit and y is the number of years. a.) Estimate the profit 7 years from now. b.) Estimate the years needed for the company to break even.

4 13. The equation for the height of a ball thrown into the air is h ( t) 16t 40t 50, where h(t) is the height of the ball after t seconds. a.) Estimate the time it takes for the ball to be 30 feet above the ground. b.) Estimate the height of the ball and seconds it takes to reach its maximum height. 14. The function f(x) = -0.0x + x + 1 models the yearly growth of a young redwood tree, f(x), in inches, with x inches of rainfall per year. a.) How many inches of rainfall per year does it take for the tree to grow 3 inches? b.) How many inches of rainfall per year results in maximum tree growth and what is the maximum yearly growth? 15. A 0-foot supporting wire is to be attached to the top of an antenna. The wire must be anchored at a distance from the base of the antenna and that distance is 4 feet more than the height of the antenna. How high is the antenna? 16. A building casts a shadow that is double the length of the building s height. If the distance from the end of the shadow to the top of the building is 300 meters, how high is the building? Express the answer in simplified radical form. Then find a decimal approximation to the nearest tenth of a meter.

5 For the quadratic functions given below, a) Express the vertex as an ordered pair. b) Is the vertex a maximum or a minimum point? c) Write the equation of the axis of symmetry. d) Find the y-intercept and express it as an ordered pair. e) Find the x-intercepts and express them as ordered pairs. f) Graph the function. 17. f(x) = -(x + 1) f(x) = -x + x + 3 y y x x 19. Rewrite the equation f(x) = x + 6x + 4 in the form f(x) = a(x-h) + k to match the equation to the graph. a. b. c. d. 0. Rewrite the equation f(x) = -x + 4x - 1 in the form f(x) = a(x-h) + k to match the equation to the graph. a. b. c. d.

6 Solve each inequality, express the answer in interval notation, and graph it on a number line. 1. x + 9x x x < 1 x 9 3. (3x-1)(x+4)(3x+6) ( x )( x 4) x