NAME: PERIOD: TABLE #: Unit 3: Quadratics (graphing, factoring, completing the square, the quadratic formula, complex numbers)

Transcription

1 NAME: PERIOD: TABLE #: Unit 3: Quadratics (graphing, factoring, completing the square, the quadratic formula, complex numbers) Textbook: Students demonstrate knowledge of how real and complex numbers are related 5.0* both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane. 6.0* Students add, subtract, multiply, and divide complex numbers. Students solve and graph quadratic equations by factoring, completing the square, 8.0* or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. Students demonstrate and explain the effect that changing a coefficient has on the 9.0* graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x b) + c. Students graph quadratic functions and determine the maxima, minima, and zeros 10.0* of the function. Date Notes Homework Quiz Score 1 U n i t 3

2 Topic: Graphs of Quadratics in Standard Form Standard: 8, 10 Standard Form of a Quadratic Function f ( x) ax bx c f(x) is like y a, b, c are just numbers the shape of the graph is a U or parabola the shape is symmetric (cut straight in the middle and both sides are the same) the point where the shape can be cut is called the vertex or the maximum or the minimum the line that cuts the parabola is called the axis of symmetry (line of symmetry) Axis of symmetry Vertex (maximum) Axis of symmetry Vertex (minimum) Symmetric property of parabolas: If you know one point on one side of the parabola, you can find one point on the other side of the parabola Is the vertex a minimum or a maximum? If the parabola opens up then the vertex is a. If the parabola opens down then the vertex is a. The axis of symmetry always crosses the. The equation for the axis of symmetry is x = U n i t 3

3 1.) Graph y 3x 4 using a table. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Axis of symmetry: Vertex: (, ) Domain: Range:.) Graph y x x 3 using a table. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Axis of symmetry: Vertex: (, ) Domain: Range: Vertex Formula or Equation for the Axis of Symmetry b x given y ax bx c a **To find the y-value for the vertex, just plug in the x-value to the original equation. 3 U n i t 3

4 3.) Graph y x x 3. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Step 1: Find the vertex. Step : Find another point on the graph. Axis of symmetry: Vertex: (, ) Domain: Range: 4.) Graph y x 4x. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Step 1: Find the vertex. Step : Find another point on the graph. Axis of symmetry: Vertex: (, ) Domain: Range: Note: Sometimes the question asks for you to find the maximum or minimum of a quadratic equation. Just find the vertex. 4 U n i t 3

5 Topic: Graphs of Quadratics in Vertex Form Standard: 8, 9, 10 The best form to graph a parabola is vertex form of a quadratic equation. y a( x h) k where (h, k) is the vertex Change the sign on the h, not the k 5.) Find the coordinates for the vertex of the following quadratic: (a) y 4( x 1) 5 (b) y ( x 3) 1 (c) y 1 ( x 6) 3 (d) y x 7 (e) y 10( x 6) Reflection over the x-axis: y x (positive in front of the x term) y x (negative in front of the x term) Narrow or Wide: y ax (a > 1 makes it narrow) y ax (0< a < 1 makes it wide) 5 U n i t 3

6 6.) Graph y ( x 1) 4. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Step 1: Find the vertex. Step : Find another point on the graph. Axis of symmetry: Vertex: (, ) Domain: Range: 7.) Graph y 1 ( x 4) 5. Label the vertex as the minimum or maximum. Find the domain and range and the equation for the axis of symmetry. Step 1: Find the vertex. Step : Find another point on the graph. Axis of symmetry: Vertex: (, ) Domain: Range: 6 U n i t 3

7 Understanding Transformations 8.) What is the equation of the function A. y ( x 3) 4 B. y ( x 4) 3 C. y ( x 4) 3 D. y ( x 3) 4 y x translated 3 units up and 4 units left? 9.) Which list of functions is ordered from widest to narrowest graph? A. 1 y 4x, y x, y x 9 3 B. 1 y x, y 4x, y x 9 3 C. 1 y x, y x, y 4x 9 3 D. 1 y 4x, y x, y x ) How does the graph of y x 3 differ from the graph of y x 4? A. The graph of y x 3 is 7 units to the right of the graph of y x 4. B. The graph of y x 3 is 7 units to the left of the graph of y x 4. C. The graph of y x 3 is 7 units below the graph of y x 4. D. The graph of y x 3 is 7 units above the graph of y x 4. Topic: Summary of Graphs Standard: 8, 9, 10 Matching activity 7 U n i t 3

8 Topic: Factoring Standard: 8 Factoring = rewrite into products For example: 1 = (3)(4) 11.) Find the greatest common factor (biggest number you can divide by evenly) (a) 4x 1 (b) 10x 38 (c) 3x 7x Factoring ax bx c b +. ac where a = 1 1.) Factor x 8x 7 13.) Factor x 6x 8 14.) Factor x 1x 3 15.) Factor x 14x ) Factor x 17x 7 17.) Factor x 6x 8 18.) Factor x 7x 1 19.) Factor x x 1 0.) Factor x 14x 3 8 U n i t 3

9 Topic: Factoring Special Cases Standard: 8 Perfect Square Trinomial a ab b ( a b)( a b) or a ab b ( a b)( a b) 1.) Factor x 14x 49.) Factor x 6x 9 3.) Factor x 10x 5 4.) Factor 4x 1x 9 5.) Factor 9x 6x 1 6.) Factor 5x 0x 4 7.) Factor 36x 84x 49 8.) **Factor 4x 1xy 9y Difference of Two Squares a b ( a b)( a b) 9.) Factor x ) Factor 5x ) Factor 36x 1 3.) Factor 64 9x 33.) ** Factor 16y 49z BINGO 9 U n i t 3

10 Topic: Factoring More Complex Quadratics Standard: 8 Factoring ax bx c Check if you can factor out something first (divide out a number) Factor by grouping 34.) Factor 5x 5x ) Factor x 7x 3 36.) Factor 3x 17x ) Factor 7x 4x 3 38.) Factor 11x 168x ) Factor 3x 54x U n i t 3

11 40.) Factor 1x 5x 7 41.) Factor 8x 18x 9 Topic: Solving Quadratics by Taking Square Roots or by Factoring Standard: 8 The zero-product property If ab = 0, then a = 0 or b = 0 or both are 0. Example: 5a = 0, so a = 0 4.) Solve x 7x ) Solve x 11x ) Find the zeros of y x 4x 6 11 U n i t 3

12 45.) Find the x-intercepts of f ( x) 16x 8x Solve by using square roots 46.) Solve 5x ) Find the x-intercepts of f ( x) 4x 5 48.) Solve 3x 4 49.) Find the solutions for y 5x ) Solve x 8x 1 U n i t 3

13 51.) Solve ( x 1) ) What are the x-intercepts for y 1 ( x 5) 4 53.) Solve 5( x 3) ) Solve ( x 7) U n i t 3

14 Topic: Complex Number i Standard: 5, 6 The number i is the number whose square is -1. i 1 and i 1 An imaginary number or complex number is any number in the form of a bi 55.) Simplify 1 Simplify ) Simplify ( 5 i) ( 4 6i) Simplify ( 3 i) (6 7i) ** Remember i 1 57.) Simplify ( 4 5i)( 7i) Simplify ( 6 i) Plotting complex numbers 58.) Plot the following on the graph below. Label each point. A 4 i B 6 + 3i C 4i D -1 E ½ -5i 14 U n i t 3

15 The complex conjugate 59.) Find the complex conjugate of each: A 4 5i B i C 4i 60.) Simplify the expression. 3 i 1 5i 61.) Simplify the expression. 1 4i 3 i 6.) Simplify the expression. 5i 5i 15 U n i t 3

16 Solving Quadratics with i 63.) Solve 4x ) Solve 5x ) Solve 4( x 1) 31 1 Topic: Completing the Square Standard: 8 Completing the square can be used for all quadratics of the form ax bx c 0. It works best when b = even number. Step 1: Divide everything by a Step : Move c to the other side. Step 3: Take half of b and square it. Add it to both sides. (to make it a perfect square) Step 4: Factor. Step 5: Square root to solve. 66.) Solve x 1x U n i t 3

17 67.) *Find the roots of y x 8x 9 68.) Find the x-intercepts of f ( x) 4x 48x 8 69.) What are the zeros of y 3x 4x 9 Topic: The Quadratic Formula Standard: 8 The Quadratic Formula: b b 4ac x for ax bx c 0. a 70.) What are the zeros of y 3x 5x using the quadratic formula? 17 U n i t 3

18 71.) Solve using the quadratic formula. x 4x 3. 7.) Find the roots for y x 3x ) Solve 0 x 3x 5 74.) Solve 0 3x x 5 18 U n i t 3

19 Topic: Unit 3 Poster Standard: 8, 9, 10 Choose one of the equations from the list below and (1) graph using any method (label x-intercepts, axis of symmetry, domain and range, vertex) () find the x-intercepts using factoring, (3) find the x-intercepts using completing the square, and (4) find the x-intercepts using the quadratic formula. Show all work for each part. Put your answers on a poster and decorate it. Provide descriptions for each step so that anyone can understand your work. The poster should be thorough and neat. You may work with one other partner on this or work by yourself. A) y x 8x 1 B) y x 4x 3 C) y x 10x 5 D) y x 6x 5 E) y x 8x 16 F) y x 1x 7 G) y x 4x 1 Graphing Quadratic Formula b x b 4ac a Complete the Square Factoring 19 U n i t 3