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4.1 Graph Quadratic Functions in Standard Form (Parabolas) Tuesday, June 08, 2010 12:31 PM Axis of symmetry Vertex Link to Parabola Animation Another Cool Animation Ch 4 Page 1 Graph: Graph: 1. 2. 3. 4. 2. Graph: 3. Graph: Graphing any quadratic function in standard form: Find To find the value of the vertex, plug in the value you found for the axis of symmetry into the original equation and solve for . Those 2 values are your vertex - intercept: Ch 4 Page 2 Graph: Graph: Maximum and Minimum values • If the graph opens up then there is a minimum value • If the graph opens down then there is a maximum value • The max or min is the coordinate of the vertex. Tell whether the function Ch 4 Page 3 has a min or max value, then find it. 4.2 Graph Quadratic Functions (Parabolas) in Vertex or Intercept Form Wednesday, June 09, 2010 8:09 AM Animated Parabola Movement Vertex Form of a Parabola: • The vertex is -- *use the opposite of h • the axis of symmetry is • opens up and opens down Graph a quadratic function in vertex form Graph: 2. Graph: 1. Graph: Ch 4 Page 4 1. 2. 3. 4. 5. Intercept Form of a Parabola: The -intercepts are and *use opposite of and The axis of symmetry is halfway between and The -coordinate of the vertex: The -coordinate is found by plugging in into the original equation and solving for . Graph: Graph: Changing an equation from intercept form to standard form Write in standard form. Multiply out the right and combine like terms. Ch 4 Page 5 Changing an equation from vertex form to standard form. Write in standard form. Write the quadratic function in standard form. 2. 1. Graph: Ch 4 Page 6 4.3 Solve x2 + bx + c = 0 by factoring Wednesday, June 09, 2010 9:10 AM Factor the expression. 1. 2. 3. 1. 2. 3. Factoring a Difference of 2 Squares: 2. 1. 3. Factoring Perfect Square Trinomials: 2. 1. Solving quadratic equations The solutions are called the roots of the equation. To solve: 1. Set the equation = 0 2. Factor completely 3. Set your factors = 0 and solve for . What are the roots of the equation: Ch 4 Page 7 4. and 3. Find roots Solve for Find zeros Find -int's Solve the equation 2. 1. 3. Zeros of a Function: In lesson 4.2 you learned that the -intercepts of the graph of are and . Because the function's value is zero when and the numbers and are also called zeros of the function. Find the zeros of the function by rewriting in intercept form. 2. 1. Ch 4 Page 8 4.4 Solve Wednesday, June 09, 2010 by Factoring 10:36 AM Factor: by grouping. You try these 2. 1. 4. 3. Factor Difference of squares and perfect square trinomials. 1. 2. 3. Ch 4 Page 9 4. Factoring out monomials(GCFs) first, then continue to factor completely. 2. 3. 1. You try these 2. 1. 3. Solve the quadratic equation by factoring and setting your factors = 0. Always start with GCF. Finding the zeros of a function by rewriting it in intercept form. Find the zeros of Ch 4 Page 10 Quiz Review Wednesday, June 09, 2010 11:22 AM Graph: Graph: Graph: Factor completely: Ch 4 Page 11 Factor completely: 1. 2. 3. 4. Solve each equation by factoring: 1. 2. 3. 4. Ch 4 Page 12 4.5 Solve Quadratic Equations by Finding Square Roots Wednesday, June 09, 2010 11:24 AM The expression is called a radical. The symbol is a radical sign, and the number beneath the radical sign is the radicand of the expression. Simplifying Square Roots • No perfect square can be left in the radical • No radicals can be left in the denominator Simplify the expression. Practice for Example 1 6. 4. 7. 2. 5. 3. Ch 4 Page 13 8. The expressions and are conjugates of each other. Their product is always a rational number Rationalizing the denominator - eliminating radicals from the denominator when simplifying. Simplify. 1. 3. 2. Simplify the expression. 4. 1. 5. 2. 6. 3. Solving a quadratic equation Ch 4 Page 14 Solving a quadratic equation 2. 1. Solve the equation. 1. 2. 3. Solve for x: Simplify: Ch 4 Page 15 4.6 Complex(Imaginary) Numbers Thursday, June 10, 2010 9:33 AM Cool Fractals are made with Complex Numbers Not all quadratic equations have real-number solutions. For example, has no realnumber solutions because the square of any real number is never a negative number. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit defined as Note that . Here are some examples Solve a quadratic equation. 1. 2. 1. 2. 3. Adding and subtracting complex numbers - same as regular variables except you must write the imaginary number last. 1. 2. 3. Multiplying complex numbers - the same as regular variables except Ch 4 Page 16 Multiplying complex numbers - the same as regular variables except You cannot leave an in your final answer! 1. 2. Dividing complex numbers - cannot leave an imaginary # in the denominator - so you must multiply by conjugate Write the quotient in standard form: Write the expression as a complex number in standard form. 1. 2. 3. 4. Ch 4 Page 17 4.7 Completing the Square Thursday, June 10, 2010 11:25 AM Solving quadratics by finding square roots. 1. Solve 2. when a =1 by completing the square. 1. Solving 2. -- when Solve: Ch 4 Page 18 -- by completing the square *must divide all terms by coefficient of x2 1. 2. Vertex Form: where is the vertex of the graph. To write a quadratic function in vertex form, use completing the square. Write: in vertex form, then identify the vertex. 1. 2. Find the maximum value of a quadratic function. Baseball: The height (in feet) of a baseball seconds after it is hit is given by this function: Find the maximum height of the baseball. Find the vertex. Ch 4 Page 19 4.8 Using Quadratic Formula to Solve Quadratic Equations Thursday, June 10, 2010 12:35 PM In order to use the quadratic formula, you must first write the equation in standard form Solve using the quadratic formula. 1. 2. 3. 4. Ch 4 Page 20 Review for test Monday, November 01, 2010 9:15 AM Solve by completing the square. 1. Simplify: 2. 3. 4. Simplify: 5. Solve for x. 6. Solve using the quadratic formula. Ch 4 Page 21 7. Solve by factoring: 8. Graph: Change into vertex form by completing the square and find the vertex. Ch 4 Page 22