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Interactive Study Guide for Students Chapter 6: Quadratic Functions and Inequalities Section 1: Graphing Q.F.’s Graph Quadratic Functions Examples A ______________ _____________ is described by an equation of the following form: __________ term f(x) = ax2 + bx + 1.Graph f(x) = 2x2 – 8x + 9 by making a table of the values. x c , where a≠0 2x2–8x+9 f(x) (x,f(x)) 0 1 __________ term ___________ term 2 The graph of a quadratic function is called a __________. 3 All parabolas have an _________ of __________. The point at which the axis of symmetry intersects the parabola is called the __________. 4 Graph of a quadratic function f(x) = ax2+ bx +c(a≠0): The y-intercept is a(0)2 + b(0) + c or ___. The equation of the axis of symmetry is x = _______ The x-coordinate of the vertex is __________. Maximum and Minimum Values 2. f(x) = x2 + 9 + 8x The y-coordinate of the vertex of the quadratic function is the _______________ value or ____________ value obtained by the function. Find the y-intercept = the eq. of the ax. Of sym.= the x-coor. of the vert.= Graph of a quadratic function f(x) = ax2+ bx +c(a≠0): Opens _____ and has a minimum value when a>0 Opens _____and has a maximum value when a<0 Make a table, including the vertex, and graph. 3. f(x) = x2 – 4x + 9 Chapter 6: Quadratic Functions and Inequalities Section 2: Solving Q.E.’s by Graphing Solve Quadratic Equations Examples When the quadratic function is set equal to a value, the result is a ___________ ___________. It can be written in the form of ax2 + bx + c = 0 where a≠0. 1. Solve x2+6x+8 by graphing. The solutions of a quadratic equation are called the ______ of the equation. One method for finding the roots of a quadratic equation is to find the ________ of the related quadratic function. The zeros of the function are the x-intercepts of its graph. These are the solutions of the related equation because f(x) = 0 at those points. A quadratic equation can have: one real solution two real solutions no real solution 2. Solve 8x – x2 = 16 by graphing. 3. Find two real numbers whose sum is 6 and whose product is 10 or show that no such numbers exist. Estimate Solutions Often, exact roots cannot be found by graphing. In this case, you can _________ solutions by stating the consecutive integers between which the roots are located. Chapter 6: Quadratic Functions and Inequalities 4. Solve –x2 + 4x – 1 = 0 by graphing. Estimate the solutions Section 3: Solving Q.E.’s by Factoring Solve Equations by Factoring Examples Graphing is one way to “solve” the quadratic equation. ___________ is another way, where you set each factor to zero. This method use the _________ _____________ Property because in each case at least one of the factors is zero . *Always check your solutions* Solve each equation by factoring. 1. x2 = 6x 2. 2x2 + 7x = 15 3. x2 - 16x + 64 = 0 4.What is the positive solution of the equation 3x2 – 3x -60 = 0? Write Quadratic Equations You have seen that a quadratic equation of the form (x-p)(x-q)= 0 has roots p and q. You can use this pattern to find a quadratic equation for a given pair of roots. Chapter 6: Quadratic Functions and Inequalities Square Root Property 5.Write a quadratic equation with ½ and -5 as its roots. Write the equation in the form ax2+bx+c==0 Section 4: Completing the Square Examples For equations like x2 – 25 = 0, you can solve by factoring. You can also use the ___________ _______ _____________ to solve such an equation: For any real number n, if x2 = n, then x = + Solve by using the Square Root Property. 1. x2 + 10x + 25 = 49 n 2. x2 -6x + 9 = 32 3. Complete the square x2 + 8x – 20 = 0 Completing the Square The Square Root Property can be used to solve quadratic equations when the side containing the q.e. is a perfect square. If it is not, use a method called ____________ the __________. 1. Find one half of b. 2. Square the result in step 1. 3. Add the result of Step 2 to x2 + bx 4. 2x2 – 5x + 3 = 0 X2 bx + (b/2)2 = (x + b/2)2 5. x2 + 4x + 11 = 0 Chapter 6: Quadratic Functions and Inequalities Section 5: The Q.F. and Discriminant Quadratic Formula Examples The exact solutions to some quadratic equations can be found by graphing, factoring, or by using the Square Root Property. While completing the square can be used to solve any quadratic equation, the process can be tedious if the equation contains fractions or decimals. Fortunately, a formula exists that can be used to solve any quadratic equation of the form ax2 + bx + c = 0. It is called the ______________ ______________. a≠0: X= Solve by using the Q.F.: 1. x2 – 12x = 28 (2 Q rts) 2. x2 + 22x + 121 = 0 rts) (1 Q ________________________ 3. 2x2 + 4x – 5 = 0 (2 I rts) Roots and the Discriminant Can you see a relationship between the value of the expression under the radical and the roots of the quadratic equation? __________________________ The expression b2 -4ac is called the ___________________. This helps you determine the number and type of roots of a quadratic equation. 2 Consider ax + bx + c = 0 Value of Discr. b2 -4ac > 0; 4ac is pef.sq. Type and # of rts. b2 - b2 -4ac > 0; b2 4ac is not pef.sq. b2 -4ac = 0 2 b -4ac < 0 2 real, rational roots 2 real, irrational roots 4. x2 – 4x = -13 (2 C rts) Describe the roots: 9x2-12x+4 = 0 Ex. of Graph 6. 2x2 + 16x + 33 = 0 7. -5x2 + 8x – 1 = 0 1 real, rational root 8. -7x + 15 x2 – 4 = 0 2 complex roots Solving Quadratic Functions 5. Method Can be Used When to Use Graphing Factoring Square Root Property Completing the Square Quadratic Formula Chapter 6: Quadratic Functions and Inequalities Section 6: Analyzing Graphs of Q.E.’s Analyze Quadratic Functions Examples A ________ of ________ is a group of graphs that displays one or more similar characteristics. How is the parent graph y=x2 similar to family graphs y=x2+ 2 and y=(x-3)2? Each function above can be written in the form y=__________________ where (_, _) is the vertex of the parabola and x = ___ is the axis of symmetry. This is called the ___________ form. Analyze the function, then draw it’s graph. 1. y = (x + 2)2 + 1 y=a(x-h)2+k h and k k h a A the values of h and k change, the graph of y=a(x-h)2+k is the graph of y=x2 translated: |h| units _____ if h is negative or |h| units ______ if h is positive, and |k| units _____ if k is positive or |k| units _______ if negative. Write Quadratic Functions in Vertex Form Complete the _______ to write the function in vertex form. k is 2. y = x2 + 8x -5 3. y = -3x2 + 6x -1 4. vertex=(-1,4), passes through (2,1) Chapter 6: Quadratic Functions and Inequalities Section 7: Graphing and Solving Q.I. Graph Quadratic Inequalities To graph a ____________ _____________ in two variables, use the same techniques used to graph linear inequalities in two variable. Graph the quadratic equation. Decide if the parabola should be solid or dashed. Test a point (x,y) inside the parabola. Check to see if the point is a solution of the inequality. If the point is a solution, shade the region _______ the parabola. If not, shade the region _________ the parabola. Examples Graph. -x2 -6x -7 1. y > Solve by graphing: Solve Quadratic Inequalities 2. x2 + 2x – 3 > 0 To solve a _____________ ________________in one variable, use the graph of the related quadratic function 3. 0 > 3x2 -7x -1 Solve a quadratic inequality algebraically, Ex: x2+x>6 Solve the related quadratic equation. Plot the solutions on a number line, using open circles: Test a value in each section to see if it satisfies the original inequality. Write the answer as a solution set. 4. A punted football is the function H(x)=4.9x2+20x+1. At what time is the ball within 5 meters from the ground?