CHAPTER 9: Quadratic Equations and Functions

Transcription

1 CHAPTER : Quadratic Equations and Functions Notes # -: Exploring Quadratic Graphs A. Graphing ax 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. Examples: x x x x The graph of a quadratic function is a U-shaped curve called a. When graphed it will look like: OR You can fold a parabola so that the two sides match exactl. This propert is called:. The highest or lowest point of the parabola is called the, which is on the axis of smmetr. B. Identifing a Vertex Identif the vertex of each graph. Tell whether it is a minimum or a maximum..).) Vertex: (, ) Vertex: (, ).).) Vertex: (, ) Vertex: (, )

2 Graph each function. State the domain, the vertex (min/max point), the range, the x-intercepts, and the axis of smmetr..) f(x)= x x -.) h(x) = x x -.) f(x) = x x Vertex: Max or min? Axis of smmetr: Vertex: Max or min? Axis of smmetr: Vertex: Max or min? Axis of smmetr:

3 .) k(x) = x + x x ) f(x) = x x x Vertex: Max or min? Axis of smmetr: Vertex: Max or min? Axis of smmetr: C. Comparing Widths of parabolas The value of a, the coefficient of the as well as the direction in which it opens. x term in a quadratic function, affects the width of the parabola When a, then the parabola is steeper, (or ) than = x When a, then the parabola is not as steep, (or ) than = x Order each group of quadratic functions from widest to narrowest graph: 0.) f ( x) x, f( x) x, f( x) x.) x, x, x

4 D. Applications.) A monke drops an banana from a branch feet above the ground. Gravit causes the banana to fall. The function ht gives the height of the banana h in feet after t seconds. a) Graph this quadratic function b) When does the banana hit the ground? t ht ( ) t (t, h(t)) 0 0 feet time (sec) ) A bungee jumper dives from a platform. The function h = -t + 0 describes her height, h, after t seconds in the air. a) What will her height be after second? b) what will her height be after seconds? c) How far did she fall between and seconds in the air?

5 Notes # -: Quadratic Functions = ax + bx + c One ke characteristic of a parabola is its vertex (min/max point). Yesterda we found the vertex after we graphed the function. It would help to find the vertex first. Vertex b find x = a plug this x value into the function (table) this point (, ) is the vertex of the parabola Graphing 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 Find the vertex of each parabola. Graph the function and find the requested information.) f(x)= x + x + a =, b =, c = x ) h(x) = x + x x Vertex: Max or min? Direction of opening? Wider or narrower than = x? Axis of smmetr: Vertex: Max or min? Direction of opening? Wider or narrower than = x? Axis of smmetr:

6 .) k(x) = x x x Vertex: Max or min? Direction of opening? Wider or narrower than = x? Axis of smmetr: Without graphing the quadratic functions, complete the requested information:.) f x x x ( ).) gx 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?.) 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? B. Application.) Suppose a particular star is projected from an aerial firework at a starting height of 0 feet with an initial upward velocit of ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation h t t time t in seconds. 0 gives the star s height h in feet at

7 Notes # -: Finding and Estimating Square Roots A. Finding Square Roots The expression means the positive, or square root. The expression means the negative square root. The expression means both the and square root Simplif each expression..).) 00.).) 0.).) 0.0 ).).) 0 0.).).). B. Rational and Irrational Square Roots Tell whether each expression is rational or irrational..).).).).) Between what two consecutive integers is.?.) Between what two consecutive integers is.?

8 .) Between what two consecutive integers is.? C. Application: Pthagorean Theorem (Review) Use the Pthagorean theorem ( ) to solve for the missing side of the right triangle. 0.).) x -: 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? )

9 B. Solving b Graphing Solve each equation b graphing the related function:.) x = 0 Find the vertex and other points on the parabola; graph. Find the x-intercepts from the graph. These are the or x ) x = 0.) x + = x x

10 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.) = 0 0.) b = 0.) (x ) =.) ( + ) =.) (r + ) =.) (m ) = 0

11 If the left side is not alread factored or squared, it!.) x + x + =.) n n + =.) w + w + =.) g + 0g + = D. Application.) A museum is planning an exhibit that will contain a large globe. The surface area of the globe will be 00 ft. Find the radius of the sphere producing this surface area. Use the equation S r, where S is the surface area and r is the radius.

12 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.) 0 x x.) x x = 0.) + + = 0.) v = 0

13 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 = ( )( ) =

14 .) The product of two consecutive integers is. Find the integers. Let x = st integer = nd integer.) The product of two consecutive odd integers is. Find the integers. Let x = st odd integer = nd odd integer.) The length of a rectangle is ft greater than its width. The area of the rectangle is ft. Find the length and the width of the rectangle. 0.) The area of a square is more square inches than there are inches in the square s perimeter. Find the length of a side of the square.

15 .) 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. + = -