CHAPTER 9: Quadratic Equations and Functions

Transcription

1 Notes # CHAPTER : Quadratic Equations and Functions -: Exploring Quadratic Graphs A. Intro to Graphs of Quadratic Equations: = ax + bx + c A is a function that can be written in the form = ax + bx + c where a, b, and c are real numbers and a 0. Ex: = x = x + = x x The graph of a quadratic function is a U-shaped curve called a. The maximum or minimum point is called the Identif the vertex of each graph; identif whether it is a minimum or a maximum..).) Vertex: (, ) Vertex: (, ).).) Vertex: (, ) Vertex: (, )

2 B. Ke Features of a Parabola: = ax + bx + c Direction of Opening: When a > 0, the parabola opens : Domain: All Real Numbers When a < 0, the parabola opens : Range: When U, then When I, then (the -coordinate of the vertex) (the -coordinate of the vertex) Width: When a <, the parabola is than When a =, the parabola is the width as When a >, the parabola is than = x = x = x Vertex: The highest or lowest point of the parabola is called the vertex Axis of smmetr: This is a vertical line passing through the vertex. Its equation is the same as the x-coordinate of the vertex x-intercepts: are the 0,, or points where the parabola crosses the x-axis. Plug in = 0 and solve for x. -intercept: is the point where the parabola crosses the -axis. Plug in x = 0 and solve for. Without graphing the quadratic functions, complete the requested information:.) f ( x) = x x +.) g( x) = x + x What is the direction of opening? What is the direction of opening? Is the vertex a max or min? Is the vertex a max or min? Wider or narrower than = x? Wider or narrower than = x?.) = x.) = x + x 0... What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? The parabola = x is graphed to the right. Note its vertex (, ) and its width. You will be asked to compare other parabolas to this graph.

3 Graph each function. Find the indicated information:.) f(x)= x x x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: 0.) h(x) = -x + x -x ) f(x) = x + x x x + x x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:

4 f x = x + x.) ( ) x x.) f(x) = x x + x x x x x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: Order each group of quadratic functions from widest to narrowest graph:.) f ( x) x, f ( x) x, f ( x) x = = =.) = x, = x, = x

5 For #, choose the graph for each given equation..) f ( x) = x x.) h( x) = x +.) g x = x x.) h( x) = x +.) ( ).) A x B x ).) C x D x

6 Notes # -: Quadratic Functions Fining the Vertex b - find x = a - plug this x-value into the function (table) - this point (, ) is the vertex of the parabola Steps for Graphing Quadratic Functions - put the vertex ou found in the center of our x- chart. - choose x-values less than and x-values more than our vertex. - plug in these x values to get more points. - graph all points & connect with a parabola. Find the vertex of each parabola. Graph the function and find the requested information.) f(x)= -x + x + a =, b =, c = Vertex: x -x + x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:.) h(x) = x + x + a =, b =, c = Vertex: x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:

7 .) k(x) = x x a =, b =, c = Vertex: x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:.) f ( x) = x x + a =, b =, c = Vertex: x x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:

8 .) f ( x) = x + x a =, b =, c = Vertex: x Y x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range:.) f ( x) = x + x a =, b =, c = Vertex: x Y x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: Without graphing the quadratic functions, complete the requested information:.) f x = x x + ( ).) g x = x + x ( ) What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x?

9 .) = x 0.) = x + x 0... What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? For #-, choose the graph for each given equation..) f ( x) = x x.) h( x) = x + x What is the direction of opening? Is the vertex a max or min? Wider or narrower than = x? h x = x + 0.) ( ) = +.) g ( x) x.) A x B x ) 0.).) C x D x

10 Notes # Simplifing Radicals To simplif a radical expression, break the number under the radical down into its & look for Example: 0 Examples: Simplif each expression as much as possible..).).).).) 0.).) 0.).) 0.)

11 Notes # -: Solving Quadratic Equations A. Solving Quadratic Equations b Graphing The solutions of a quadratic equation and the related x-intercepts are often called of the equation or of the function..) The function f(x) = x + x is graphed to the left. a) Circle and name the zeros of the function graphed here. (, ) and (, ) b) Use this graph to solve the equation: x + x = 0 (This is asking: At what x-values does = 0? ) B. Solving b Graphing Solve each equation b graphing the related function: Find the vertex and other points on the parabola; graph. Find the x-intercepts from the graph. These are the or..) x = x

12 .) x = 0.) x + = x x C. Solving Quadratic Equations Using Square Roots Isolate the variable or expression being squared (get it ) Square root both sides of the equation (include + and on the right side!) This means ou have equations to solve!! Solve for the variable (make sure there are no roots in the denominator).) x =.) x =.) x = 0.) m = 0

13 .) = 0 0.) b = 0.) (x ) =.) ( + ) =.) (r + ) =.) (m ) = 0 If the left side is not alread factored or squared, it!.) x + x + =.) n n + =.) w + w + =.) g + 0g + =

14 For #-, choose the graph for each given equation..) f ( x) = x x 0.) h( x) = x x.).) h( x) = x +.) ( ) g x = x + x 0.) A x B x ).) C x D x

15 Notes # -: Solving Quadratic Equations b Factoring A. Solving Quadratic Equations Zero Product Propert List some pairs of numbers that multipl to zero: ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 ( )( ) = 0 What did ou notice? Use this pattern to solve for the variable:. get = 0 and factor (sometimes this is done for ou). set each ( ) = 0 (this means to write two new equations). solve for the variable (ou sometimes get more than solution).) ()(x) = 0.) ()(x + ) = 0.) -( ) = 0.) (m + )(m ) = 0.) ( ) 0 w w =.) x x + = 0.) x x = 0.) + + = 0.) v = 0

16 0.) x + x = 0.) x x =.) x + x =.) v(v + ) = 0.) b(b ) = (b + ) B. Solving Word Problems with Quadratics Steps:. Draw a picture and define our variable (let statement). Write an equation. Get = 0 (bring all variables and numbers to one side). Factor completel and solve. Do all the answers make sense?. Write our answer in a complete sentence Translate and solve:.) The square of a positive number minus twice the number is. Find the number. Let n = - =.) One more than a negative number times one less than that number is. Find the number. Let n = ( )( ) =

17 .) Two less than the square of a number is equal to the number. Find the number. - =.) The sum of the square of a number and three times the number is the same as one less than the number. Find the number. + = - C. Review of Solving b Square Roots Isolate the expression being squared (ou ma need to factor first) Square root both sides of the equation (include + and equations!) Solve for the variable and simplif (reduce radical, rationalize denominator, etc) Solve using the square root method. (Check to make sure ou have two solutions!).) x = 0.) ( x + ) =

18 .) m + =.) g + =.) x x + =.) x x + =

19 Notes # -: Completing the Square So far in this course, we have solved quadratics b, and. We will eventuall learn two more was to solve quadratics. Solve these equations. What makes these quadratics eas to solve? a) (x ) = b) (k + ) = Solving quadratics b helps us turn all quadratics into this form. Complete the square: Take half the b (the x coefficient) Square this number (no decimals leave as a fraction!) Add this number to the expression Factor it should be a binomial, squared ( ).) x + x +.) m m + ( )( ) ( )

20 Find the value of n such that each expression is a perfect square trinomial:.) w + w + n.) k k + n.) j j + n.) + + n Solving b Completing the Square: Collect variables on the left, numbers on the right Divide ALL terms b a; leave as fractions (no decimals!!) Take half of the b term. Square this number and add it to BOTH sides Factor the side with the variables into a perfect square. Square root both sides (include a and equation!) Set up equations (one set equal to a positive number & the other set equal to a negative number) Solve for the variable (simplif all roots).) x + x = 0.) x x = 0.) k k = 0 0.) m m + = 0

21 .) + = 0.) x x = 0 Cumulative Review: Solving Quadratics Solve #- b a method of our choice:.) k k =.) m = 0 Solve b using square roots:.) w =.) = 0.) m = 0.) (x ) = 0 Solve b completing the square:.) x 0x + = 0.) m + m = 0

22 .) = 0 0.) x 0x = -0.) x + x + = 0.) ax + bx + c = 0

23 Notes #0 : Using the Quadratic Formula A. Review of Simplifing Radicals and Fractions Simplif expression under the radical sign, reduce Reduce onl from ALL terms of the fraction.) ±.) ± 0.) ± 0.) ±.) ± ( ) ()()().) ± () ( )( ) B. Solving Quadratics using the Quadratic Formula So far, we have solved quadratics b: (), (), (), and () The final method for solving quadratics is to use the quadratic formula. Solving using the quadratic formula: Put into standard form (ax + bx + c = 0) List a =, b =, c = Plug a, b, and c into Simplif all roots, reduce ± x = b b ac a

24 Solve b using the quadratic formula:.) x + x = x = ± b b ac a (std. form): a = b = c =.) x x = - (std. form): x = ± b b ac a a = b = c =.) x = x.) x = 0x

25 .) -x + x = -.) x = x Review of Solving Quadratics: Solve b factoring:.) m +m = 0.) x x = 0 Solve b using square roots:.) b = 0 0.) = 0.) (x + ) = Solve b completing the square:.) x 0x + = 0.) x x = 0

26 .) m + m + = 0.) + = 0 Solve b using the quadratic formula:.) x 0 = 0.) x x + = 0

27 Notes # : Using the Discriminant Quadratic equations can have two, one, or no solutions. You can determine how man solutions a quadratic equation has before ou solve it b using the. The discriminant is the expression under the radical in the quadratic formula: b ± b ac x = a Discriminant = b ac If b ac = 0, then the equation has solution/x-intercept If b ac < 0, then the equation has 0 real solutions/x-intercepts If b ac > 0, then the equation has solutions/x-intercepts Finding the number of x-intercepts Determine whether the graphs intersect the x-axis in zero, one, or two points..) x x = +.) = x x 0 A. Finding the number of solutions Find the number of solutions for the following:.) x x =.) x = x.) x x =.) x = x +

28 C. Review of Solving Quadratics Solve b factoring:.) x + x = -0.) (x ) = x(x + ) Solve b using square roots:.) b = 0.) (x + ) = Solve b completing the square:.) x 0x = 0.) x x = 0

29 Solve using the quadratic formula:.) x + x =.) x = x Graph the quadratic. Name the vertex, axis of smmetr, x-intercepts, domain, and range..) f(x)= x a =, b =, c = Vertex: x Y x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: x-intercepts :

30 Notes #: Chapter Review Solve b factoring:.) x x = 0.) x + x = Solve b square rooting:.) = Solve b completing the square: Solve using the quadratic formula:.) x x = 0.) x x = 0 Find the discriminant; find the Define a variable, write an equation, number of real solutions: and solve:.) x x + = 0.) Three more than a positive number times two less than that positive number is. Find the number. Find the vertex of each parabola. Graph the function and find the requested information.) f(x)= x x a =, b =, c = Vertex: X Y x Vertex: Direction of opening? Max or min? Width compared to = x? Axis of smmetr: Domain: Range: x-intercepts :