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Chapter 2 Study Guide 2.1 Using Transformations to Graph Quadratic Functions The graph of a quadratic function is a parabola. A parabola is a curve shaped like the letter U. f(x) a(x h)2 k(a 0) Quadratic function You can make a table to graph a quadratic function. Graph f(x) x2 3. x f (x) x2 4x 3 (x, f (x)) 0 f(0) 02 4(0) 3 3 (0, 3) 1 f(1) 12 4(1) 3 0 (1, 0) 2 f(2) 22 4(2) 3 1 (2, 1) 3 f(3) 32 4(3) 3 0 (3, 0) 4 f(4) 42 4(4) 3 3 (4, 3) Vertex Form of Quadratic Function Horizontal and vertical translations change the vertex of f(x) x2. Parent Function Transformation f(x) x2 g(x) (x h)2 k Vertex: (0, 0) Vertex: (h, k) The vertex of g(x) (x 4)2 2 is (4, 2). The graph of f(x) x2 is shifted 4 units right and 2 units down. 2.2 Properties of Quadratic Functions in Standard Form You can use the properties of a parabola to graph a quadratic function in standard form: f(x) ax2 bx c, a 0. Property Example: f(x) x2 2x 2 a 0: opens upward a 0: opens downward a 1, b 2, c 2 a 0, so parabola opens downward. Axis of symmetry: x b , 2a Vertex: b 2a b f 2a Axis of symmetry: x b ( 2) 1 2a 2( 1) b f f ( 1) 1( 1)2 2( 1) 2 3 2a Vertex: (1, 3) y-intercept: c To graph f(x) x2 2x 2: y-intercept is 2, so (0, 2) is a point on the graph. 2.3 Solving Quadratic Equations by Graphing and Factoring Solve the equation ax2 bx c 0 to find the roots of the equation. Find the roots of x2 2x 15 0 to find the zeros of f x x2 2x 15. x2 2x 15 0 Factor, then multiply x 5x 3 0 to check. x 5 0 or x 3 0 Solve each equation for x. x 5 or x 3 Set each factor equal to 0. To check the roots, substitute each root into the original equation: Equation: x2 2x 15 0 x2 2x 15 0 Root: x 5 x3 Check: 52 25 15 32 23 15 25 10 15 0 9 6 15 0 The roots of x2 2x 15 0 are 5 and 3. The roots of the equation The zeros of f x x2 2x 15 are 5 and 3. are the zeros of the function. Some quadratic equations have special factors. Difference of Two Squares: a2 b2 a b a b Perfect Square Trinomials: a2 2ab b2 a b2 a2 2ab b2 a b2 Always write a quadratic equation in standard form before factoring. 16x2 25 16x2 25 0 16 and 25 are perfect squares. Use the difference of two squares to factor. 4x2 52 0 4x 54x 5 0 4x 5 0 or 4x 5 0 x 5 5 or x 4 4 Try to factor a perfect square trinomial if the coefficient of x and the constant term are perfect squares. 4x2 12x 9 0 2x2 22x3 32 0 4x2 and 9 are perfect squares. 2x 32x 3 2x 32 0 2x 3 0 x 3 2 The factors are the same. 2.4 Completing the Square You can use the square root property to solve some quadratic equations. Square Root Property To solve x2 a, take the square root of both sides of the equation. Solve Remember: 22 4, and 22 4. x2 a x2 a x a 4x 5 43. 4x2 48 Add 5 to both sides. x2 12 Divide both sides by 4. x 2 12 Take the square root of both sides. x 12 Simplify. Think: x 2 3 Solve The coefficient of x2 should be 1 to use the square root property. 2 x2 12x 36 50. x 62 50 Factor the perfect square trinomial. x 6 2 Take the square root of both sides. 50 x 6 50 Subtract 6 from both sides. x 6 50 Simplify. Think: x 6 5 2 You can use a process called completing the square to rewrite a quadratic of the form x2 bx as a perfect square trinomial. To complete the square of x2 bx, 2 b add . 2 Think: Multiply the coefficient of x by 1 . Then square it. 2 2 b b x bx x 2 2 2 2 Complete the square: x2 8x ?. Step 1 Identify b, the coefficient of x: b 8. Step 2 b Find : 2 Step 3 b Add : 2 Step 4 Factor: 2 2 2 2 b 8 4 16 2 2 2 x 2 8 x 16 x2 8x 16 x 42 Check: x 42 x 4x 4 x2 8x 16 Use as a factor. 2.5 Complex Numbers and Roots An imaginary number is the square root of a negative number. Use the definition 1 i to simplify square roots. Simplify. 25 25 1 Factor out 1. 25 Separate roots. 1 5 1 Simplify. 5i Express in terms of i. Imaginary Real Complex numbers are numbers that can be written in the form a bi. The complex conjugate of a bi is a bi. Write as a bi Find 0 5i 5i The complex conjugate of 5i is 5i. You can use the square root property and imaginary solutions. 1 i to solve quadratic equations with Solve x2 64. x 2 64 x 8i Take the square root of both sides. Express in terms of i. Check each root: 8i2 64i 2 641 64 8i2 64i 2 641 64 Remember: Solve 5x2 80 0. 5x2 80 Subtract 80 from both sides. x 16 Divide both sides by 5. x 2 16 Take the square root of both sides. 2 x 4i Express in terms of i. Check each root: 54i 2 80 54i 2 80 516i 2 80 516i 2 80 801 80 801 80 0 0 2.6 The Quadratic Formula The Quadratic Formula is another way to find the roots of a quadratic equation or the zeros of a quadratic function. Find the zeros of f x x2 6x 11. Step 1 Set f x 0. Step 2 Write the Quadratic Formula. x Step 3 Substitute values for a, b, and c into the Quadratic Formula. x2 6x 11 0 b b 2 4ac 2a a 1, b 6, c 11 b b2 4ac 6 x 2a Step 4 2 Simplify. x Step 5 6 4 1 11 2 1 6 6 4 1 11 6 2 1 2 36 44 6 80 2 2 Write in simplest form. x 6 80 3 2 80 3 2 16 5 2 3 4 5 32 5 2 Remember to divide both terms of the numerator by 2 to simplify. The discriminant of ax2 bx c 0 a 0 is b2 4ac. Use the discriminant to determine the number of roots of a quadratic equation. A quadratic equation can have 2 real solutions, 1 real solution, or 2 complex solutions. Find the type and number of solutions. x2 10x 25 2x2 5x 3 3x2 4x 2 Write the equation in standard form: Write the equation in standard form: Write the equation in standard form: 2x2 5x 3 0 x2 10x 25 0 3x2 4x 2 0 a 2, b 5, c 3 a 1, b 10, c 25 a 3, b 4, c 2 Evaluate the discriminant: Evaluate the discriminant: Evaluate the discriminant: b2 4ac b2 4ac b2 4ac 52 423 102 4125 42 432 25 24 100 100 16 24 49 0 8 When b2 4ac 0, the equation has 2 real solutions. When b2 4ac 0, the equation has 1 real solution. When b2 4ac 0, the equation has 2 complex solutions. 2.7 Solving Quadratic Inequalities Graphing quadratic inequalities is similar to graphing linear inequalities. Graph y x2 2x 3. Draw the graph of y x2 2x 3. • a 1, so the parabola opens downward. Step 1 • vertex at (1, 4) b 2 1 , and f 1 4 2a 2 1 • y-intercept is 3, so the curve also passes through (2, 3) Draw a solid boundary line for or . (Draw a dashed boundary line for or .) Step 2 Shade below the boundary of the parabola for or . (Shade above the boundary for or .) Step 3 Check using a test point in the shaded region. Use 0, 0. y x2 2x 3 ?: 0 02 20 3 :03 2.8 Curve Fitting with Quadratic Models When the second differences are constant in a pattern of data, the data could represent a quadratic function. x 2 3 4 5 6 y 6 12 20 30 42 Check that the x-values are equally spaced. Find the first differences. This means the differences between successive y-values. 12–6 20–12 30–20 42–30 6 8 10 12 Find the second differences. This means the differences between successive first differences. 8–6 10–8 12–10 2 2 2 Data Set 1 is a quadratic function. Second differences are constant. They are all 2. 2.9 Operations with Complex Numbers Graphing complex numbers is like graphing real numbers. The real axis corresponds to the x-axis and the imaginary axis corresponds to the y-axis. To find the absolute value of a complex number, use a bi a2 b2 . 7i 3 i 0 2 7 2 49 Think: 7i 0 7i; so a 0 and b 7. 3 2 1 2 9 1 Think: 3 i 3 1i; so a 3 and b 1. 10 7 To add or subtract complex numbers, add the real parts and then add the imaginary parts. 4 i 2 6i Remember to distribute when subtracting. Then group to add the real parts and the imaginary parts. 4 i 2 6i 4 2 i 6i 6 7i Use the Distributive Property to multiply complex numbers. Remember that i 2 1. 3i 2 i 6i 3i 2 Distribute. 6i 31 Use i 2 1. 3 6i Write in the form a bi. 4 2i 5 i 20 4i 10i 2i 2 Multiply. 20 6i 21 Combine imaginary parts and use i 2 1. 22 6i Combine real parts.