Slide 1 / 200 Quadratic Functions Table of Contents Slide 2 / 200 Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots Solve Quadratic Equations by Completing the Square Solve Quadratic Equations by Using the Quadratic Formula The Discriminant Solving Non-Quadratics Solving Rational Equations Solving Radical Equations Quadratic & Rational Inequalities Slide 3 / 200 Key Terms Return to Table of Contents Slide 4 / 200 Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves Completing the square: Adding a term to x2 + bx to form a trinomial that is a perfect square Discriminant: b2 - 4ac in a quadratic in standard form Slide 5 / 200 Maximum: The y-value of the vertex if a < 0 and the parabola opens downward Minimum: The y-value of the vertex Max if a > 0 and the parabola opens upward Min Parabola: The curve result of graphing a quadratic equation Slide 6 / 200 Quadratic Equation: An equation that can be written in the standard form ax2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0. Quadratic Function: Any function that can be written in the form y = ax2 + bx + c. Where a, b and c are real numbers and a does not equal 0. Vertex: The highest or lowest point on a parabola. Zero of a Function: An x value that makes the function equal zero. Slide 7 / 200 Identifying Quadratic Functions Return to Table of Contents Slide 8 / 200 Any function that can be written in the form y = ax2 + bx + c Where a, b, and c are real numbers and a ≠ 0 Examples Question: Is 2x2 = x + 4 a quadratic equation? Answer: Yes Question: Is 3x - 4 = x + 1 a quadratic equation? Answer: No Slide 9 / 200 Explain Characteristics of Quadratic Functions Return to Table of Contents Slide 10 / 200 A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0. 2 The form ax + bx + c = 0 is called the standard form of the quadratic equation. The standard form is not unique. For example, x2 - x + 1 = 0 can be written as the equivalent equation -x2 + x - 1 = 0. 2 Also, 4x - 2x + 2 = 0 can be written as the equivalent equation 2x2 - x + 1 = 0. Slide 11 / 200 Practice writing quadratic equations in standard form: (Reduce if possible.) Write 2x2 = x + 4 in standard form: 2x2 - x - 4 = 0 Slide 12 / 200 Write 3x = -x2 + 7 in standard form: x2 + 3x - 7 = 0 Slide 13 / 200 Write 6x2 - 6x = 12 in standard form: 2 x - x - 2 = 0 Slide 14 / 200 Write 3x - 2 = 5x in standard form: Not a quadratic equation Slide 15 / 200 Slide 16 / 200 2. The graph of a quadratic is a parabola, a u-shaped figure. 3. The parabola from a polynomial function will open upward or downward. Slide 17 / 200 4. A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point. Slide 18 / 200 5. The domain of a quadratic function is all real numbers. 6. To determine the range of a quadratic function, ask yourself Slide 19 / 200 two questions: Is the vertex a minimum or maximum? What is the y-value of the vertex? If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value. The range of this quadratic is -6 to Slide 20 / 200 If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value. The range of this quadratic is to 10 Slide 21 / 200 7. An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form Slide 22 / 200 8. The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic function will have two, one or no real x-intercepts. Slide 23 / 200 1 True or False: The vertex is the highest or lowest value on the parabola. True False Slide 24 / 200 2 If a parabola opens upward then... A a>0 B a<0 C a=0 Slide 25 / 200 3 The vertical line that divides a parabola into two symmetrical halves is called... A discriminant B perfect square C axis of symmetry D vertex E slice Slide 26 / 200 Quadratic Equations Finding Zeros (x- intercepts) Slide 27 / 200 Solve Quadratic Equations by Graphing Return to Table of Contents Slide 28 / 200 Vocabulary Every quadratic function has a related quadratic equation. A quadratic equation is used to find the zeroes of a quadratic function. When a function intersects the x-axis its y-value is zero. When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax2 + bx + c 0 = ax2 + bx + c ax2 + bx + c = 0 Slide 29 / 200 One way to solve a quadratic equation in standard form is find the zeros of the related function by graphing. A zero is the point at which the parabola intersects the x-axis. A quadratic may have one, two or no zeros. Slide 30 / 200 How many zeros do the parabolas have? What are the values of the zeros? No zeroes 2 zeroes; 1 zero; x = -1 and x=3 x=1 Slide 31 / 200 One way to solve a quadratic equation in standard form is to find the zeros or x-intercepts of the related function. Solve a quadratic equation by graphing: Step 1 - Write the related function. Step 2 - Graph the related function. Step 3 - Find the zeros (or x intercepts) of the related function. Slide 32 / 200 Step 1 - Write the Related Function 2x2 - 18 = 0 2x2 - 18 = y y = 2x2 + 0x - 18 Slide 33 / 200 Step 2 - Graph the Function 2 y = 2x + 0x – 18 The vertex is found by substituting the Use the same six step process for graphing x-coordinate of the Axis of Vertex Symmetry into the equation and solving for y. The axis of symmetry is x = 0. The vertex is (0,-18). Find the y intercept -- It is -18. Find two other points (2,-10) and (3,0) Two Points The point where the line passes through the y-axis. This occurs when the x-value is 0. Y Intercept Y Slide 34 / 200 Step 2 - Graph the Function y = 2x2 + 0x – 18 Graph the points and reflect them across the axis of symmetry. x = 0 # # (3,0) # # (2,-10) # (0,-18) Slide 35 / 200 Step 3 - Find the zeros y = 2x2 + 0x – 18 Solve the equation by graphing the related function. # x = 0 # The zeros appear (3,0) to be 3 and -3. # # (2,-10) # (0,-18) Slide 36 / 200 Step 3 - Find the zeros y = 2x2 + 0x – 18 Substitute 3 and -3 for x in the quadratic equation. Check 2x2 – 18 = 0 2 2(3) – 18 = 0 2(-3)2 – 18 = 0 2(9) - 18 = 0 2(9) - 18 = 0 18 - 18 = 0 18 - 18 = 0 0 = 0 ü 0 = 0 ü The zeros are 3 and -3. Slide 37 / 200 4 Solve the equation by graphing the related function. -12x + 18 = -2x2 Step 1: What of these is the related function? 2 A y = -2x - 12x + 18 B y = 2x2 - 12x - 18 2 C y = -2x + 12x - 18 Slide 38 / 200 5 What is the axis of symmetry? y = -2x2 + 12x - 18 A -3 Formula: -b 2a B 3 C 4 D -5 Slide 39 / 200 y = -2x2 + 12x - 18 6 What is the vertex? A (3,0) B (-3,0) C (4,0) D (-5,0) Slide 40 / 200 y = -2x2 + 12x - 18 7 What is the y- intercept? A (0,0) B (0, 18) C (0, -18) D (0, 12) Slide 41 / 200 2 8 y = -2x + 12x - 18 If two other points are (5,-8) and (4,-2), what does the graph look like? A B Slide 42 / 200 9 If two other points are (5,-8) and (4,-2), what does the graph of y = -2x2 + 12x - 18 look like? A B C D Slide 43 / 200 10 y = -2x2 + 12x - 18 What is(are) the zero(s)? A -18 B 4 C 3 D -8 Slide 44 / 200 Solve Quadratic Equations by Factoring Return to Table of Contents Solving Quadratic Equations Slide 45 / 200 by Factoring Review of factoring - To factor a quadratic trinomial of the form x2 + bx + c, find two factors of c whose sum is b. Example - To factor x2 + 9x + 18, look for factors whose sum is 9. Factors of 18 Sum 1 and 18 19 2 and 9 11 x2 + 9x + 18 = (x + 3)(x + 6) 3 and 6 9 Slide 46 / 200 When c is positive, its factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive. When b is negative, the factors are negative. Slide 47 / 200 Remember the FOIL method for multiplying binomials 1. Multiply the First terms (x + 3)(x + 2) x x = x2 2. Multiply the Outer terms (x + 3)(x + 2) x 2 = 2x 3. Multiply the Inner terms (x + 3)(x + 2) 3 x = 3x 4.