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Quadratics
- 1. Algebra 1 QUADRATIC FUNCTIONS • Quadratic Expressions, Rectangles and Squares • Absolute Value, Square Roots and Quadratic Equations • The Graph Translation Theorem • Graphing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 • Completing the Square • Fitting a Quadratic Model to Data • The Quadratic Formula • Analyzing Solutions to Quadratic Equations • Solving Quadratic Equations and Inequalities
- 2. Algebra 2 QUADRATIC FUNCTIONS Quadratic – quadratus (Latin) , ‘to make square’ 𝑎𝑥2 + 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 f 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 − 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 Standard form of a quadratic: 𝑎𝑥2 + 𝑏𝑥 + 𝑐
- 3. Algebra 3 QUADRATIC FUNCTIONS Quadratic expressions from Rectangles and Squares Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is w meters wide, write the total area of the pool and walkway in standard form. Write the area of the square with sides of length 𝑥 + y in standard form
- 4. Algebra 4 QUADRATIC FUNCTIONS Binomial Square Theorem For all real numbers x and y, 𝑥 + 𝑦 2 = 𝑥2 + 2xy + 𝑦2 𝑥 − 𝑦 2 = 𝑥2 − 2xy + 𝑦2 Note: When discussing this, ask students whether any real-number values of the variable give a negative value to the expression. [ The square of any real number is nonnegative].
- 5. Algebra 5 QUADRATIC FUNCTIONS Challenge Have students give quadratic expressions for the areas described below. 1. The largest possible circle inside a square whose side is x. 2. The largest possible square inside a circle whose radius is x.
- 6. Algebra 6 QUADRATIC FUNCTIONS Absolute Value, Square Roots and Quadratic Equations In Geometry • 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑛 𝑓𝑟𝑜𝑚 0 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒 In Algebra • 𝑥 = 𝑥, 𝑓𝑜𝑟 𝑥 ≥ 0 −𝑥, 𝑓𝑜𝑟 𝑥 < 0
- 7. Algebra 7 QUADRATIC FUNCTIONS Activity 1. Evaluate each of the following. 42, −4 2, 9.32, −9.3 2 2. Find a value of x that is a solution to 𝑥2 = 𝑥. 3. Find a value of x that is not a solution to 𝑥2 = 𝑥.
- 8. Algebra 8 QUADRATIC FUNCTIONS Absolute Value – Square Root Theorem For all real numbers x, 𝑥2 = 𝑥 Example 1 Solve 𝑥2 = 40 Example 2 A square and a circle have the same area. The square has side 10. What is the radius of the circle?
- 9. Algebra 9 QUADRATIC FUNCTIONS Challenge The Existence of Irrational Numbers Prove that 2 cannot be written as a simple fraction.
- 10. Algebra 10 QUADRATIC FUNCTIONS Graphs and Translations Consider the graphs of 𝑦1 = 𝑥2 and 𝑦2 = 𝑥 − 8 2 What transformation maps the graph of the first function onto the graph of the second? Graph – Translation Theorem In a relation described by a sentence in x and y, the following two processes yield the same graph: 1. replacing 𝑥 by 𝑥 − ℎ and 𝑦 by 𝑦 − 𝑘 2. applying the translation 𝑇ℎ,𝑘 to the graph of the original relation.
- 12. Algebra 12 QUADRATIC FUNCTIONS Example 1 Find an equation for the image of the graph of 𝑦 = 𝑥 under the translation 𝑇5,−3. Corollary The image of the parabola 𝑦 = 𝑎𝑥2 under the translation 𝑇ℎ,𝑘 is the parabola with the equation 𝑦 − 𝑘 = 𝑎 𝑥 − ℎ 2 Example 2 a. Sketch the graph of 𝑦 − 7 = 3 𝑥 − 6 2 b. Give the coordinates of the vertex of the parabola c. Tell whether the parabola opens up or down d. Give the equation for the axis of symmetry.
- 13. Algebra 13 QUADRATIC FUNCTIONS Graphing 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 Suppose ℎ = −16𝑡2 + 44𝑡 + 5 a. Find ℎ when 𝑡 = 0, 1, 2 𝑎𝑛𝑑 3 b. Explain what each pair 𝑡, ℎ tells you about the height of the ball. c. Graph the pairs 𝑡, ℎ over the domain of the function. Note: Two natural questions about the thrown ball are related to questions about this parabola. 1. How high does the ball get? The largest possible value of h. 2. When does the ball hit the ground?
- 14. Algebra 14 QUADRATIC FUNCTIONS Newton’s Formula ℎ = − 1 2 𝑔𝑡2 + 𝑣0 𝑡 + ℎ0 • 𝑔 is a constant measuring the acceleration due to gravity • 𝑣0 is the initial upward velocity • ℎ0 is the initial height • the equation represents the height ℎ of the ball off the ground at time 𝑡.
- 15. Algebra 15 QUADRATIC FUNCTIONS Completing the Square 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 𝑦 − 𝑘 = 𝑎 𝑥 − ℎ 2 𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚 Completing the square geometrically and algebraically Theorem To complete the square on 𝑥2 + 𝑏𝑥, 𝑎𝑑𝑑 1 2 𝑏 2 . 𝑥2 + 𝑏𝑥 + 1 2 𝑏 2 = 𝑥 + 1 2 𝑏 2 = 𝑥 + 𝑏 2 2
- 16. Algebra 16 QUADRATIC FUNCTIONS Practice • Going back to ℎ = −16𝑡2 + 60𝑡 + 5, find the maximum height of the ball. • Rewrite the equation 𝑦 = 𝑥2 + 10𝑥 + 8 in vertex form. Locate the vertex of the parabola. • Suppose 𝑓 𝑥 = 3𝑥2 + 12𝑥 + 16 a. What is the domain of 𝑓? b. What is the vertex of the graph? c. What is the range of 𝑓?
- 17. Algebra 17 QUADRATIC FUNCTIONS Fitting a Model to Data
- 18. Algebra 18 QUADRATIC FUNCTIONS Practice The number of handshakes ℎ needed for everyone in a group of 𝑛 people, 𝑛 ≥ 2, to shake the hands of every other person is a quadratic function of 𝑛. Find three points of the function relating ℎ and 𝑛. Use these points to find a formula for this function.
- 20. Algebra 20 QUADRATIC FUNCTIONS The Angry Blue Bird Problem What if Blue Bird’s flight path is described by the function ℎ 𝑥 = −0.005𝑥2 + 2𝑥 + 3.5 Where is Blue Bird when she’s 8 feet high?
- 21. Algebra 21 QUADRATIC FUNCTIONS The Quadratic Formula If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑎𝑛𝑑 𝑎 ≠ 0 , 𝑡ℎ𝑒𝑛 𝑥 = −𝑏± 𝑏2−4𝑎𝑐 2𝑎 Challenge How was it derived?
- 22. Algebra 22 QUADRATIC FUNCTIONS Practice Solve 3𝑥2 + 11𝑥 − 4 = 0 The 3-4-5 right triangle has sides which are consecutive integers. Are there any other right triangles with this property? Challenge: Find a number such that 1 less than the number divided by the reciprocal of the number is equal to 1.
- 23. Algebra 23 QUADRATIC FUNCTIONS How Many Real Solutions Does a Quadratic Equation Have? Discriminant Theorem Suppose 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 are real numbers with 𝑎 ≠ 0. Then the equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has i. two real solutions if 𝑏2 − 4𝑎𝑐 > 0. ii. one real solution if 𝑏2 − 4𝑎𝑐 = 0. iii. two complex conjugate solutions if 𝑏2 − 4𝑎𝑐 < 0.
- 24. Algebra 24 QUADRATIC FUNCTIONS Practice Determine the nature of the roots of the following equations. Then solve. a. 4𝑥2 − 12𝑥 + 9 = 0 b. 2𝑥2 + 3𝑥 + 4 = 0 c. 2𝑥2 − 3𝑥 − 9 = 0
- 25. Algebra 25 QUADRATIC FUNCTIONS Solving Quadratic Equations Extracting Square Roots Factoring Completing the Square Quadratic Formula
- 26. Algebra 26 QUADRATIC FUNCTIONS Factoring Steps: • Transform the quadratic equation into standard form if necessary. • Factor the quadratic expression. • Apply the zero product property by setting each factor of the quadratic expression equal to 0. Zero Product Property – If the product of two real numbers is zero, then either of the two is equal to zero or both numbers are equal to zero. • Solve each resulting equation. • Check the values of the variable obtained by substituting each in the original equation.
- 27. Algebra 27 QUADRATIC FUNCTIONS Practice Solve the following equations. 1. 𝑥2 + 7𝑥 = 0 2. 𝑥2 + 9𝑥 = −8 3. 4𝑥2 − 81 = 0 4. 4𝑥2 + 9 = 12𝑥 5. 𝑥 − 1 2 − 5 𝑥 − 1 + 6 = 0
- 30. Algebra 30 QUADRATIC FUNCTIONS HOW TO SOLVE? 1. find the "=0" points 2. in between the "=0" points, are intervals that are either greater than zero (>0), or less than zero (<0) 3. then pick a test value to find out which it is (>0 or <0)
- 32. Algebra 32 QUADRATIC FUNCTIONS Here is the plot of 𝑦 = 𝑥2 − 𝑥 − 6 The equation equals zero at -2 and 3 The inequality "<0" is true between -2 and 3.
- 33. Algebra 33 QUADRATIC FUNCTIONS Practice 1. Find the solution set of 𝑥2 − 5𝑥 − 14 > 0. 2. Graph 𝑦 > 𝑥2 − 5𝑥 − 14 3. A stuntman will jump off a 20 m building. A high-speed camera is ready to film him between 15 m and 10 m above the ground. When should the camera film him?