May 16, Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials. Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0

Transcription

1 Aim: To review for Quadratic Function Exam #2 Homework: Study Review Materials Warm Up - Solve using factoring: 5x 2 + 7x + 2 = 0

2 Review Topic Index 1. Consecutive Integer Word Problems 2. Pythagorean Theorem Word Problems 3. Finding zeroes/roots on calculator 4. Discriminant 5. Completing the Square 6. Writing Equation from Roots 7. Linear Quadratic Systems 8. Graphing Linear Quadratic System Word Problem

3 Concept Review: (1) Consecutive Integer Word Problems Consecutive integers are 1 apart. x x + 1 x + 2 Consecutive odd or even integers are 2 apart. x x + 2 x + 4 Product means to multiply together. Example If there are THREE consecutive odd integers, the product of the two SMALLEST consecutive odd integers can be written as:

4 Sample Question: (1) Consecutive Integer Word Problems 1 A The ages of three babies are consecutive even integers. If the product of the two oldest babies is 100, which can be used to find their ages? x 2 + 4x = 0 B C D x 2 + 3x - 98 = 0 (x + 2) + (x + 4) = 100 2x + 6 = 100

5 Concept Review: (2) Pythagorean Theorem Word Problems Pythagorean Theorem: In a right triangle, a 2 + b 2 = c 2 c must be hypotenuse (opposite right angle) Strategy: Draw a right triangle. Label one side x. Label other sides in terms of x. Substitute into Pythagorean Theorem and solve using quadratic strategies. Ex: In a right triangle, the hypotenuse is 2 longer than one leg and 7 longer than the other. Find the side lengths. x-2 x x-7

6 Sample Question: (2) Pythagorean Theorem Word Problems In a right triangle, one leg is 4 longer than the other. And the hypotenuse is 8 longer than the shorter leg. Find the dimensions of the right triangle algebraically.

7 Concept Review: (3) Finding Roots on a Calculator To find roots on a calculator, use: 2nd - Calc - Zeros Left Bound Right Bound Left and Right Bound should be placed correctly. Guess should be placed anywhere near the root. Right Bound Left Bound If you can't see the graph nicely: Zoom - Standard or Zoom - Fit or Zoom - Out or

8 Sample Question: (1) Finding Roots on a Calculator 2 The zeros of the function g(x) = 4x x - 84 A B C D 7 and -3-7 and 3-2 and and -100

9 Sample Question: (2) Finding Roots on a Calculator 3 The zeros of the function g(x) = x 3-25x A B C D -5-5 and 5-5 and 0-5, 0, and 5

10 Concept Review: (4) The Discriminant The discriminant is found in the quadratic formula It describes how many roots there are and tells you about the roots. b 2-4ac > 0 no real roots b 2-4ac < 0 no real roots b 2-4ac = 0 1 real root If perfect square, roots are rational. If not, roots are irrational.

11 Sample Question: (4) Discriminant Calculate the discriminant for the function f(x) = -2x 2 + 5x - 3 How many real roots does f(x) have?

12 Sample Question: (4) Discriminant Calculate the discriminant for the function g(x) = 11x x - 1 How many real roots does f(x) have? Are they rational or irrational? How could you check your solution using your calculator?

13 Concept Review: (5) Completing the Square Standard Form: f(x) = Ax 2 + Bx + C Vertex Form: f(x) = a(x - h) 2 + k Steps for CTS: 1. Make a = 1 (Divide every term by "a") 2. Move constant (c) over 3. Add (b/2) 2 to both sides to create perfect binomial 4. Move constant/values back over to make into "f(x)" form Be Careful: Don't forget about the value you divided out in step 1. Since this isn't an equation equal to 0, but rather a function, it does not simply go away and must be preserved. Helpful hint: When in vertex form, the sign of the number in the parentheses will match the sign of "B" in standard form.

14 Sample Question: (5) Completing the Square Solve the following quadratic using completing the square. x 2 + 8x - 2 = 0

15 Sample Question: (5) Completing the Square Solve the following quadratic using completing the square. x 2 + x - 5 = 0

16 Sample Question: (5) Completing the Square Which of the following could be one of the steps to solve the equation x 2-5x + 1 = 0 a) (x + 5/2) 2 = 21/4 b) (x + 5) 2 = 21/4 c) (x - 5) 2 = 21/4 d) (x - 5/2) 2 = 21/4

17 Concept Review: (6) Writing the Equation from Roots a b Let's make a function with roots a and b. Roots make the output, f(x), equal to zero. So, we want to design an equation so that f(x) becomes zero when we replace x with a or b. Which one will work? f(x) = (x - a)(x - b) OR f(x) = (x + a)(x + b)

18 Sample Question: (1) Writing an Equation from Roots 4 Create a function with roots -4 and -3. Be sure to expand any binomials.

19 Sample Question: (2) Writing an Equation from Roots 5 Write a function with roots equal to 8, 0, and -8. Be sure to expand any product of binomials.

20 Concept Review: (7) Quad-Linear System of Equations Quad: y = ax 2 + bx + c Linear y = mx + b Could have 0, 1, or 2 solutions. The solutions to the system are the points that make both equations true. So, where the parabola and line intersect. To solve algebraically: Get y alone in both equations Set equal to each other Move all terms to one side (set equal to 0) Solve using any method of solving quadratics.

21 Sample Question: (7) Quad-Linear Systems Ex 1: Find the solutions to the system below algebraically f(x) = 3x 2 + 4x - 7 g(x) = 2x + 1

22 Sample Question: (7) Quad-Linear Systems Bob is solving a system of equations and arrives at the step (2x + 1)(x - 5) = 0. Which of the following could represent the original system? a) f(x) = 2x 2-4x - 7 b) f(x) = 2x 2 + 4x - 9 g(x) = 5x - 2 g(x) = 3x - 4

23 Sample Question: (7) Quad-Linear Systems Ex 1: Find the solutions to the system below algebraically f(x) = x 2 + 8x - 12 g(x) = - 2x + 12

24 Concept Review: (8) Graphing Linear Quadratic System Word Problems Output (Temperature, $, etc.) The - coordinate of this point tells how long it took for the output of one graph to equal the output of the other. Steps: Solve the system. Find the - coordinate. This distance tells how much higher the output of one graph is compared to the other. Steps: Find the outputs. Subtract them. Time

25 Sample Question: (1) Graphing Linear Quadratic System Word Problems 6 The height, h, of a baseball in meters at t seconds after it is tossed out of a window is modeled by: h = -5t t A bird flies toward the baseball. The trajectory of the bird is given by the equation h = 3t + 3. Will the bird hit the baseball? If so, when? At 2 seconds, how much higher will the baseball be compared to the bird?

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