Unit Calendar. Date Sect. Topic Homework HW On-Time Apr , 2, 3 Quadratic Equations & Page 638: 3-11 Page 647: 3-29, odd

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1 Name/Period: Unit Calendar Date Sect. Topic Homework HW On-Time Apr , 2, 3 Quadratic Equations & Page 638: 3-11 Graphs Page 647: 3-29, odd Apr Solving Quadratic Equations by Factoring Solving Quadratic Equations by Square Roots Apr. 8 (B) Quiz # 1 Curve of Best Fit Apr. 12 (B) Solving Quadratic Equations using the Quadratic Formula Practical Applications of Quadratic Equations Page 578: 3-13, odd Page 586: 20-38, even Page 597: 23-37, odd Page 655: 3-13, odd Quiz #1 Study Guide Complete Page 12 in the Notes Complete Page 14 in the Notes Quiz #2 Study Guide Apr. 14 (B) Quiz #2 SOL Review SOL Practice Problems #1 Page 1 of 14

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3 Unit 10 Key Concepts Standard Form of a Quadratic Function: The Zeros (Roots): Properties of Equality: Quadratic Formula: xx = bb ± bb22 44aaaa 22aa Falling Object Model: hh = 1111tt 22 + ss h = ending height t = the time (in seconds) that the object falls s = the starting height (in feet) when the object is dropped Page 3 of 14

4 Quadratic Graphs and Their Properties A quadratic function is a function that can be written in the form y = ax 2 + bx + c, where a 0. This form is the standard form of a quadratic function. A quadratic function is a type of nonlinear function that models certain situations where the rate of change is not constant. What are some examples? The graph of a quadratic function is a symmetric curve with a highest or lowest point corresponding to a maximum or minimum value (the vertex). The graph is called a parabola. The parabola opens up if a is positive, and down if a is negative. y = x 2 y = -x 2 The solutions for a quadratic equation are the x-intercepts of the graph. The solutions are also called the roots or zeros of the function because they are where y = 0. There can be one real solution, two real solutions, or no real solution, depending on where, or if, the parabola crosses the x-axis. y = x 2 y = x 2 2 y = x one real solution two real solutions no real solution Page 4 of 14

5 Axis of Symmetry the line that divides the graph so that either side is a mirror image of the other. The equation for the line is: xx = bb 2222 Vertex the point that is the maximum or minimum value of the graph. Find the x-coordinate and then substitute for x to find y using: xx = bb 2222 You can solve quadratic equations by graphing the function on a graphing calculator to find where the graph intersects the x-axis (the zeros). Use the table on the calculator to find the values of x where y = 0 (the zeros). Practice. a) Find the zeros. b) Find the Axis of Symmetry c) Find the Vertex yy = xx 2 + 5xx + 6 yy = xx 2 xx 12 yy = xx 2 7xx + 10 yy = 2xx 2 8xx 3 yy = 3xx 2 + 6xx + 2 ff(xx) = xx 2 + 2xx 5 Page 5 of 14

6 Solving Quadratic Equations by Factoring The solutions to a quadratic equation are the roots or the zeros. This is where y = 0. Therefore, to solve a quadratic equation, y = ax 2 + bx + c, we set y = 0 and solve for: 0 = ax 2 + bx + c In Unit 10, we learned to factor. To solve a factored quadratic, we find where the product of our factors is zero. The only way that a product can be zero is if at least one of the factors is zero. For example, if xy = 0, then x and/or y must = 0. Examples. Solve the equation. (xx 5)(xx + 1) = 0 xx(xx + 3) = 0 3(xx 4)(xx 8) = 0 (xx + 4) 2 = 0 xx 2 3xx + 54 = 0 3xx xx + 24 = 0 Page 6 of 14

7 Sometimes, you have to solve for 0 to put the equation in standard form. xx xx = 10 4xx 2 27xx = 40 Practice. Solve the equation. xx xx + 18 = 0 xx 2 + 6xx 9 = 0 6xx 2 2xx = 0 xx 2 + 7xx = 0 2xx 2 5xx 3 = 0 5xx 2 + 7xx = 2 Page 7 of 14

8 Solving Quadratic Equations Using Square Roots You can solve equations of the form x 2 = k by finding the square roots of each side. For example, the solutions of xx 2 = 81 are ± 81 or ±9. How do you know you can solve using square roots? The equation has an x 2 - term and a constant term, but no x-term. So, you can write your equation in the form x 2 = k and then find the square roots of each side. Examples. What are the solutions to the equations? 3xx 2 75 = 0 mm 2 36 = 0 3xx = 0 4dd = 16 Practice. Solve the equations. xx 2 = 25 xx 2 = 49 xx 2 36 = 0 xx 2 81 = 0 xx 2 2 = xx2 = 50 Page 8 of 14

9 Solving Quadratic Equations Using the Quadratic Formula Remember: Quadratic equations can have two, one, or no real number solutions. A quadratic equation can never have more than two solutions. Remember: Quadratic equations are solved by setting y = 0. You can find the solution(s) of any quadratic equation using the quadratic formula. If aaxx 22 + bbbb + cc = 00, and aa 0, then xx = bb ± bb22 44aaaa 22aa The quadratic formula not provide a real number solution if: WHY? aa = 0 or bb 2 4aaaa < 0 Examples. Solve using the quadratic formula. Find the roots of: xx 2 + 7xx + 12 = 0 xx 2 10xx + 25 = 0 Page 9 of 14

10 Solve using the quadratic formula. 2xx 2 3xx 5 = 0 Find the x-intercepts of: 3xx 2 2xx = 6 Practice. Solve/Find the Roots/Zeros/x-intercepts. Round to the nearest hundredths, when necessary. yy = xx 2 + 3xx 9 xx 2 8 = 2xx xx 2 4xx = xx xx + 2 = 0 3xx 2 11xx + 4 = 0 7xx 2 2xx = 8 Page 10 of 14

11 Curve of Best Fit The relationship between the variables can be determined by creating and analyzing the curve of best fit. The curve of best fit is a curve that has the same number of data elements above and below it and follows the trend of the graph. Sample Data Set A Set B Set C Set D x (L1) y (L2) x (L1) y (L2) x (L1) y (L2) x (L1) y (L2) Enter Data into Calculator: 1) STAT, Edit (1) 2) Enter data in L1 (x values) and L2 (y values). Make sure the data is accurate and aligned. 3) STATPlot (2 nd + Y=). With cursor on Plot 1, ENTER. Make sure On is highlighted, and that xlist: L1 and ylist: L2 4) ZOOM, ZoomStat (9) 5) This displays the Scatter Plot. Calculating Line of Best Fit using data that you have already entered. 1) STAT. Toggle to CALC and select LineReg (ax+b) (4). 2) Enter the list where you entered your data. The default is L1,L2 so if that is where your data is stored, you do not need to change anything. 3) Scroll to Calculated and ENTER. 4) This is your line of best fit in Slope-intercept form: y = ax + b a = slope b = y-intercept Calculating Curve of Best Fit using data that you have already entered. 1) STAT. Toggle to CALC and select QuadReg (5). 2) Enter the list where you entered your data. The default is L1,L2 so if that is where your data is stored, you do not need to change anything. 3) Scroll to Calculated and ENTER. 4) This is your curve of best fit in Standard Form: y = ax 2 + bx + c Page 11 of 14

12 Clear your data lists each time: STAT, 4, 2 nd, 1, comma, 2 nd, 2, ENTER Find the Line of Best Fit and Curve of Best Fit fit for each of the four data sets. Data Set A Data Set B Line of Best Fit Line of Best Fit a = a = b = b = Curve of Best Fit Curve of Best Fit a = a = b = b = c = c = Data Set C Data Set D Line of Best Fit Line of Best Fit a = a = b = b = Curve of Best Fit Curve of Best Fit a = a = b = b = c = c = Page 12 of 14

13 Practical Uses of Quadratic Equations Falling Object Model: When something is dropped, you can estimate its distance from the ground using this model: hh = 1111tt 22 + ss h = ending height t = the time (in seconds) that the object falls s = the starting height (in feet) when the object is dropped Examples. How long does it take for an object that is dropped from 300 feet to hit the ground? How long does it take for a free-fall ride at an amusement park to drop 121 feet? The goal of an egg dropping contest is to create a container so that an egg can be dropped 32 feet without breaking. How long will it take to hit the ground? Page 13 of 14

14 Practice (Complete for Homework). 1) A boulder falls from a cliff in a storm. The cliff is 70 feet high. How long will it take for the boulder to hit the road below? 2) An object is dropped from a 56-foot bridge over a bay. How long will it take for the object to reach the water? 3) A ball is dropped from a six-floor window at a height of 70 feet. When will the ball hit the ground? 4) The light from the top of a 240-foot tall observation tower falls. How long will it take for the light to hit the ground? 5) An object falls from the top of a 100-foot pole. After how much time will the object hit the ground? Page 14 of 14