Chapter 2. Linear and Quadratic Function

Transcription

1 Chapter. Linear and Quadratic Function.1 Properties of Linear Functions and Linear Models.8 Equations and Inequalities Involving the Absolute Value.3 Quadratic Functions and Their Zeros.4 Properties of Quadratic Functions.6 Building Quadratic Models.7 Complex Zeros of a Quadratic Function

2 .1 Properties of Linear Fn Slope of a Line The slope of the line is for two different points (x 1, y 1 ) and (x, y ) is m = y x y x 1 1 = vertical change horizontal change =Average Rate of Changes

3 .1 Ways to generate the linear Equation Slope Intercept Form: y = mx + b Point Slope Form: y y 1 = m( x x 1 )

4 .1 Ways to write the linear equation Slope Intercept Form: y = mx + b Standard Form (General Form): ax + by = c

5 .1 Equations of lines A point-slope form of an equation of the line through (x 1, y 1 ) with slope m is y - y 1 = m( x - x 1 ) This formula is from the slope, y x y y x y = = m ( x ) m x 1 Which is same as y y = ( ) mx 1 x 1

6 .1 Determine whether the given fn is linear or nonlinear. A. x y B. x y

7 .1 Find the slope and y-intercept of the following equation a. y = -3(x 1). b. x - 4y = -1 c. 3 x y = 1 4

8 .1The Straight Line Depreciation: Ex. If a $30,000 machine depreciates 8% of its original value each year, answer the following question. a. Slope? b. Find a function that expresses the machine s book value V after t years have elapsed c. Graph the linear equation. d. When will be the book value of $8,400?

9 EX. A town s population has been growing linearly. In 000, the population was 50,000 and the population has been growing by 3000 people each year. Write an equation for the population, P, as the function of t years.

10 .8 Equations and Inequalities Involving the Absolute Value Function We note that the graph of f intersects in two points with the line y=6. A) What is the solution of the system? B) Solve algebraically. 8 y= f(x)= x

11 .8 A) Find the solution for the system graphically. B) Find the solution algebraically x-3 >

12 .8 For x-4 < 4, Find the solution A) Graphically B) Algebraically

13 .3 Quadratic Functions and their zero Quadratic function is written as where a, b and c are constants and ( x) ax bx c f = + + a 0 Given the quadratic function, a. Find the y intercept. f ( ) x = x + x+ 6 b. Find the x intercept. c. Find the vertex point. d. Draw the graph.

14 Derive the quadratic formula by.3 the completing square! How do we start?

15 .3 A Quadratic Formula b ± b 4ac x = a

16 Find the roots or zeros of following functions. a. b. f f ( x) = x 4x 1 ( x) = x 8x+ 16 c. f( x) =x 1

17 .3 What are the intercepts of the graph of the function or real zeros of each function? a. f ( x) = x+ x 0 b. f ( x) = (1 x) 6(1 x) 16

18 .3 Two fishing boats depart a harbor at the same time, one traveling east, the other south. The eastbound boat travels at a speed 3 mi/h faster than the southbound boat. After two hours the boats are 30 mi apart. Find the speed of the southbound boat.

19 .4 Properties of quadratic functions ( x) = ax bx c f + + If a leading coefficient a is positive, then the graph is a concave upward. If a leading coefficient a is negative, then the graph is a concave downward. The graph, a parabola, is symmetric to the axis of symmetry. If a>0, then the vertex value of y is Min. If a<0, then the vertex value of y is Max. b b The vertex point is (, f( ) ) a a

20 ( x) = ax bx c f + + A quadratic function can be expressed in the standard form ( x) a( x h) k f = + by completing square. The graph of a parabola with vertex (h, k); the parabola opens upward if a > 0 or downward if a < 0.

21 .4 Write in standard form the following function and specify the vertex point. a. f ( x) = x 6x 1 b. f ( ) 4 x = x + x 1 3 3

22 Minimum Value of a Quadratic Fn Given the quadratic function: f ( x) = 5x 30x+ 49 A)Express f in standard form B)Sketch the graph of f C)Find the minimum value of f

23 Maximum Value of a Quadratic Fn Given the quadratic function: f ( x) = x + x+ A)Express f in standard form B)Sketch the graph of f C)Find the maximum value of f

24 .4Vertex is (-1, 3) and y intercept = - a. Find the quadratic function. b. Determine where fn is increasing and where it is decreasing. c. Determine the domain and the range of function.

25 .6 Quadratic Models A soft-drink vendor analyzes his sales records, and finds that if he sells x cans of soda in one day, his profit in dollars is given by: P( x) = 0.001x + 3x 1800 A)How many cans must he sell for maximum profit? B)What is his maximum profit per day?

26 A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time is given by: h( t) = 4.9t + 4t + 8 A)From what height was the ball thrown? B)How high above ground does the ball get at its peak? C)When does the ball hit the ground?

27 EX. A farmer has 3000 meters of fencing and he wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

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29 .7 Complex Zeros of a quadratic function Complex number = Real number + Imaginary number = a + bi x = b± b 4ac a When do we get complex roots?

30 Complex roots occur when b 4ac< 0 Find the roots and draw the graph for the following: Discriminant = b 4ac a) f ( x) =x + 4 b) f ( x) = x 4x + 1 c) f( x) = 3x + 6x+ 4

31 Determining the characteristic of the solution of a quadratic equation by using the discriminant. a) f ( x) = 4x 1x + 9 b) f ( x) = 9x 6x + 1 c) f ( x) = x x + 8