Unit 5 Test: 9.1 Quadratic Graphs and Their Properties Quadratic Equation: (Also called PARABOLAS) 1. of the STANDARD form y = ax 2 + bx + c 2. a, b, c are all real numbers and a 0 3. Always have an x 2 How do we determine which way it opens? -if leading coefficient a is positive, it opens up Ex: y = 3x 2 +2x -5 - if leading coefficient a is negative it opens down Ex: y = -4x 2 +3x -10 Maximum/Minimum Point (x, y) Depends on opening It is the VERTEX Axis of Symmetry vertical line drawn through middle of graph, you can fold one half onto other b USE: x = to state the EQUATION. 2a FINDING VERTEX AND AXIS OF SYMMETRY Label what a, b and c are equal too Axis of Symmetry: (x = #) Vertex: (x, y) b b 1) Plug into x = 1) Plug into x = 2a 2a 2) Simplify the right side 2) Simplify the right side 3) Leave answer as an equation 3) Plug into the quadratic to find y 4) Write answer as (x, y) Example: For the following, state the vertex, axis of symmetry and if it s a max or min value. 1) y = 2x 2 + 4x - 6 2) y = x 2 6 1
3) 4) y = -3x 2 6x + 2 5) y = -2x 2 6) y = -2x 2 + 2x - 3 2
9.2: Graphing Quadratic Functions Steps to Graph 1) List out a, b and c 2) Find the vertex 3) Choose x values above the vertex and below it 4) Make a t-chart and fill 5) Graph the points Examples: Graph the following quadratic functions. State the axis of symmetry and the coordinates of the vertex. Graph the axis of symmetry along with the quadratic. 1) y = -3x 2 + 12x + 1 2) f(x) = 2x 2 + 4x - 1 y y x x 3
Story Problem Applications: During Halftime of a basketball game, a sling shot launches T-shirts in the crowd hitting spectators in the head. A t-shirt has a velocity of 72 ft/s. It is caught 35 ft above the court. How long will it take the t-shirt to reach its maximum height? What is the maximum height? The function h = -16t 2 +72t + 5 gives the t-shirts height after t seconds. Suppose a tennis player hits a ball over the net. The ball leaves the racket 0.5m above the ground. The equation h = -4.9t 2 + 3.8t + 0.5 gives the ball s height h in meters after t seconds. When will the ball be at its highest point in this path? Rounds your answer to the nearest tenth? 4
9.3: Solving Quadratic Equations: Day 1 WHAT DOES THIS ALL MEAN?? Roots/Solutions/X-intercepts are all the same thing.where it crosses the x-axis!!!! What are all the ways to solve quadratics? 1) Solve by graphing. (Only works if they cross at specific points.) 2) Solve by factoring. (Must factor by grouping and not everything is factorable.) 3) Solve by graphing calculator. (Only applies if they cross at specific points.) 4) Solve by square roots. (Only works is there is no x term or you factor to something squared) 5) QUADRATIC FORMULA (Works for every situation) 5
Square Root Property If you isolate the x 2, you can use this to solve. Remember x 2 = 9 means what number times itself equals 9. YOU MUST INCLUDE POSITIVE AND NEGATIVE NUMBERS!!!! Example: x 2 81 = 0 x 2 = 81 Isolate the x 2 x = Take the square root of each side Remember to include positive and negative Examples: Solve the following quadratics. 1) x 2 = 64 2) w 2 36 = 64 3) 3x 2 + 12 = 0 4) (x - 4) 2 = 25 Examples: Find the roots of the following quadratics. Round to the nearest 10 th. 1) x 2 25 = 0 2) x 2 + 6x -7 = 0 3) x 2 + 9 = 0 6
9.4: Solve Quadratics: Day 2 Graphing works best when the graph is given to you. Square roots work best when you have no bx term. Examples: What are the solutions of each equation? 1) (t 5)(t + 7) = 0 2) (2x + 3)(x 4) = 0 3) x(x + 4) = 0 Using factoring to solve equations Steps: 1) Make the equation equal to zero 2) Make certain it is in the form x 2 + bx + c (Standard form of a quadratic) 3) Factor the trinomial after getting it to equal zero (Factor by Grouping) 4) Apply the zero product property from above and solve 7
Examples: Solve the following and check you solutions. Round to the nearest 10 th. 4) x 2 + 5x + 6 = 0 5) x 2-6x = - 8 6) 4x 2-21x - 18 = 0 7) x 2 = 4x Story Problem: 8) You are constructing a frame for the rectangular photo shown. You want the frame to be the same width all the way around and the total area of the frame and photo should be 315in 2. What should the outer dimensions of the frame be? 8
9.4: Solve Quadratics: Day 3 Examples: What are the solutions of the following? Round to the nearest 10 th. 1) x 2-6x = 247 2) x 2 + 6x = 216 3) x 2 + 9x + 15 = 0 4) x 2 14x + 16 = 0 **5) 3x 2 + 8x 96 = 0 Example: You are planning a flower garden consisting of 3 square plots surrounded by a 1-ft border. The total area of the garden and the border is 100 ft 2. What is the side length of each square plot? 9
10.2: Simplifying Radicals Examples: Simplify the following. 1) 2) 3) 4) Factor Tree Factor down completely Circle pairs of numbers Simplifying Radicals Biggest Perfect Square -- Find pairs of factors --Use product property Examples: Find the simplest form of the following expressions. 5) 6) 10
7) 8) 9) 10) #1 Rule: NO RADICAL IS ALLOWED IN THE DENOMINATOR!!!! You Must Rationalize to Remove it. Multiply top and bottom by the radical in the bottom and simplify. Examples: Simplify the following expressions. 11) 10 5 12) 2 6 13) 12 y 27 14) 11
9.6: The Quadratic Formula and Using The Discriminant - used to solve any quadratics or when you can t factor - slow down and work out every step - quadratic must be in standard form: ax 2 + bx + c = 0. Discriminant: Tells how many real solutions there are Solving using the formula: 1) Make sure it is in standard form 2) List out what a, b, and c all equal 3) Rewrite the formula using the values listed above 4) SLOW DOWN AND SOLVE FROM THE INSIDE OUT Examples: State the value of the discriminant and determine the number of real roots. 1) 2x 2 + 10x + 11 = 0 2) 4t 2 20t = -25 3) 3x 2 + 4x = -2 12
Examples: What are the solutions to the following quadratics. 4) x 2 2x 8 = 0 5) x 2 4x = 21 6) 2x 2 3x = -5 7) 4x 2 + 12x + 9 = 0 ALL OF THESE METHODS WILL SOLVE QUADRATICS. THE QUADRATIC FORMULA WILL WORK IN ANY SITUATION!!! 13