"The state of your life is nothing more than a reflection of your state of mind." ~~ Dr. Wayne W. Dyer Quadratics

Algebra 2 Units One Variable Analysis Linear Functions Linear Systems Polynomial Functions Radical and Rational Functions Quadratics Exponential Functions Trigonometric Functions Conic Sections

Quadratic Standards

Quadratic Regression [STAT] "Edit..." option. You should then see a table. Across the top should be L1, L2, and L3. If not, see below. If there is any data under L1 or L2, move the selection so you have the list highlighted, press the [CLEAR] button, and then press the down arrow to clear the list. When both lists are cleared, select the empty data in L1 by moving your selection over the "_ _ _ _ _ _". Start entering your X-values under L1. Make sure you enter the data in order. When you are done entering the X- values, move to the L2 column and enter your Y-values. Make sure they are also in order. 2nd QUIT return to the main screen. Hit STAT to return again to the Statistics menu. Hit the right-arrow key to go to CALC. At this point, you have several choices. LinReg will give you a line that best fits your points. QuadReg will give you a quadratic function, aka a second-order polynomial. Choose the one you want and hit ENTER. The calculator does not graph your curve for you, but it does tell you what the curve is. For instance, if I run a QuadReg on the data above, the calculator gives me: This tells me that the best quadratic fit for my data is the curve y=1.33x 2 -7x Record the a, b, c and r 2

Quadratics Debrief State the domain and range in words and as sets. BaseHeightPerimeterArea

Quadratics Debrief Is this relationship a function? Justify. BaseHeightPerimeterArea

Quadratics Debrief Would it be appropriate to connect the scatter plot points to form a continuous graph for this situation? What if the rectangle were built out of thread, rather than tiles? BaseHeightPerimeterArea

Quadratics Debrief Write an equation which generalizes the pattern, using your given value for P. –Find the equation using algebra. Area = Base (Perimeter -2(Base)) F(x) = x (P – 2x) F(x) = Px – 2x 2 F(x) = 24x – 2x 2 –Compare your equation to the Quadratic Regression Equation on the Calculator. –r 2 should be close to 1

Beginning of Class 1. Make sure you use the restroom and bring all materials to class. (Textbook, Pencil, Notebook, Paper) 2. Greet everyone as you enter classroom. 3. Sit in your assigned seat 4.Check Board for Assignments due and Upcoming Events 5. Have materials easily accessible. 7. Start bellwork. You will have approx 5 minutes to complete the assignment Everyone should be seated and working when the bell rings. 6. Place homework on desk for review Mrs. Motlow Classroom Procedures

Vertical Projectile Motion h(t) = -1/2 gt^2 + vt + h Where g = gravity (9.8 m/sec) t = time h = height v = initial velocity h(t) = height for given time

The Ball Toss Experiment A good data set for the ball being tossed up produces a graph in the shape of what figure? –Parabola How does this shape describe the motion of the object that was tossed? –Distance from sensor over time Find the vertex of the parabola. –Highest point (x, y) What is the value of the maximum height achieved by the object? –Y value of the vertex How long did it take for the object to return to its starting position? –Final time – initial time What was the height of the object after 1.5 seconds? –?

The Ball Toss Experiment Fit a quadratic function to the data that was collected. (Use Quad Reg on your calculator) –-4.84x x – 4.8 (r 2 =.93) Graph the function found in number 7 on the same screen with the scatter plot of collected data. How good is the fit? –View side by side or check the correlation coefficient 10. The values of a, b, and c correspond to the acceleration due to gravity, the initial velocity of the object, and the initial height of the object. Are the values that you found in # 7 reasonable? Explain. –h(t) = -4.9t^2 + vt + h

The Ball Toss Experiment -4.84x x – 4.8 Calculate Maximum Height H(t) = -4.84x x – 4.8 (max at t – 1.1) h(1.1) =.7506 Find Roots (solution) 0 = (.686, 1.46) 0 = -4.84(.686) (.686) – =.036 (rounding) 0 = -4.84(1.46) (1.46) – = (rounding)

Quadratic Equations y = ax 2 + bx + c Parabola Roots y = 0 Solutions to Equation Vertex- Lowest/Highest Point Axis of Symmetry = -b/(2a) x of the Vertex Domain – All Numbers Range – y Value at Vertex

Quadratic Equations y = x 2 Parabola Roots x = 0 Vertex- Lowest/Highest Point (0,0) Axis of Symmetry = -b/(2a) = -0/2 = 0 x=0 Domain – All Numbers Range – y ≥ 0 Root, x = 0

Graph Quadratic Equation y = 2x 2 -4x - 5 Minimum Axis of Symmetry x = -b/(2a) x = -(-4)/(2(2))=1 Vertex (1, -7) Roots- -.85, 2.8 Domain – All Numbers Range – y ≥ -7

Graph Quadratic Equation y = -x 2 +4x - 1 Axis of Symmetry Vertex Maximum Axis of Symmetry x = -b/(2a) x = -4/(2(-1))=2 - a, opens down Vertex (2, 3) Roots-.2, 3.7 Domain – All Numbers Range – y ≤ 3

Graph Quadratic Equation y = -3x 2 -6x + 4 Axis of Symmetry Vertex Maximum Axis of Symmetry x = -b/(2a) x = 6/(2(-3))=-1 - a, opens down Vertex (-1, 7) Roots- -2.5,.5 Domain – All Numbers Range – y ≤ 7

Two Real Solutions We can see that the zeros of the function are –4 and –2. Answer: The solutions of the equation are –4 and –2. CheckCheck the solutions by substituting each solution into the original equation to see if it is satisfied. x 2 + 6x + 8 = 0 0 = 0 (–4) 2 + 6(–4) + 8 = 0 (–2) 2 + 6(–2) + 8 = 0 ??

Solve x 2 + 2x – 3 = 0 by graphing. A.B. C.D. –3, 1–1, 3 –3, 1–1, 3

A.0 and 1, 3 and 4 B.0 and 1 C.3 and 4 D.–1 and 0, 2 and 3 Solve x 2 – 4x + 2 = 0 by graphing. What are the consecutive integers between which the roots are located?

A.about 6 seconds B.about 7 seconds C.about 8 seconds D.about 10 seconds HOOVER DAM One of the largest dams in the United States is the Hoover Dam on the Colorado River, which was built during the Great Depression. The dam is feet tall. Suppose a marble is dropped over the railing from a height of 6 feet above the top of the dam. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h(t) = –16t 2 + h 0, where t is the time in seconds and h 0 is the initial height above the water in feet.

Summary Quadratic Equations y = ax 2 + bx + c Parabola Roots y = 0, Solution Vertex- Lowest/Highest Point Axis of Symmetry = -b/(2a) x of the Vertex Domain – All Numbers Range – y Value at Vertex

Working with the person next to you Complete the following problems from the book Page , 23, 33, 41 Page 263, ,