ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola The U-shaped graph of a quadratic function Vertex The lowest or highest point on a parabola Axis of symmetry The vertical line that divides the parabola into mirror images and passes through the vertex Minimum and maximum value For y= ax 2 + bx+ c, the vertex's y-coordinate is the minimum value of the function if a 0 and its maximum value if a 0. PARENT FUNCTION FOR QUADRATIC FUNCTIONS The parent function for the family of all quadratic functions is f(x) = x 2. The graph is shown below The lowest or highest point on a parabola is the vertex for f(x) = x2 is (0,0) The axis of symmetry divides the parabola into mirror images and passes through the vertex For f(x) = ax 2, and for any quadratic function g(x)= ax 2 + bx + c where b = 0, the vertex lies on the y-axis and the axis of symmetry is x = _0_.
Example 1 Graph a function of the form y = ax 2 + c Graph y = 2x 2 + 2. Compare the graph with the graph of y = x 2. PROPERTIES OF THE GRAPH OF y = ax 2 + bx + c Characteristics of the graph of y = ax 2 + bx + c: The graph opens up if a _ _ 0 and opens down if a _ _ 0. The graph is narrower than the graph of y = x 2 if a _ _ 1 and wider if a _ _ 1. b b The axis of symmetry is x = and the vertex has x-coordinate. 2 a 2 a The y-intercept is _c_. So, the point (0, _c_) is on the parabola. Example 2 Graph a function of the form y = ax 2 + bx + c Graph y = x 2 + 4x 3.
MINIMUM AND MAXIMUM VALUES Words For y = ax 2 + bx + c, the vertex's y-coordinate is the minimum value of the function if a _ _ 0 and the maximum value if a _ _ 0. Example 3 Find the minimum or maximum value Tell whether the function y = 3x 2 + 12x 6 has a minimum value or a maximum value. Then find the minimum or maximum value. Checkpoint Complete the following exercises. Graph the function. Label the vertex and axis of symmetry 1. y = x 2 2. y = x 2 4x + 2 3. Find the minimum value of y = 2x 2 6x + 6
4.2 Graph Quadratic Functions in Vertex or Intercept Form Goal Graph quadratic functions in vertex form or intercept form. VOCABULARY Vertex form A quadratic function written in the form y = a(x h) 2 + k Intercept form A quadratic function written in the form y = a(x p)(x q) GRAPH OF VERTEX FORM y = a(x h) 2 + k The graph of y = a(x h) 2 + k is the parabola y = ax 2 translated _horizontally_ h units and _vertically_ k units. The vertex is (_h_, _k_ ). The axis of symmetry is x = _h_. The graph opens up if a _ _ 0 and down if a _ _ 0. Example 1 Graph a quadratic function in vertex form and find the maximum or minimum value. 1 Graph y = (x + l) 2 2. 2 GRAPH OF INTERCEPT FORM y = a(x p)(x q): Characteristics of the graph y = a(x p)(x q): The x-intercepts are _p_ and _q_. The axis of symmetry is halfway between ( _p, 0) and ( _q_, 0). It has equation x = p q 2 The graph opens up if a _>_ 0 and opens down if a _<_ 0.
Example 2 Graph a quadratic function in intercept form and find the maximum or minimum value. Graph y = 2(x 1)(x 5). Checkpoint Complete the following exercises. 1. Graph the function. Label the vertex and the axis of symmetry and find the maximum or minimum value. y = (x 3) 2 + 4 2. Graph the function. Label the vertex, axis of symmetry, and the x-intercepts and find the maximum or minimum value. y = (x 4)(x + 2)
FOIL METHOD Words To multiply two expressions that each contain two terms, add the products of the _First_ terms, the _Outer_ terms, the _Inner_ terms, and the _Last_ terms. Example F O I L (x + 4)(x + 7) = x 2 + 7x + 4x + 28 = x 2 + 11x + 28 Example 3 Change from intercept form to standard form Write y = 3(x + 2)(x 5) in standard form. Example 4 Change from vertex form to standard form Write f(x) = 5(x + 2) 2 + 8 in standard form. Checkpoint Write the quadratic function in standard form. 3. y = 4(x 3) 2 10 4. y = 3(x 7)(x + 6)
4.3 Solve x 2 bx c 0 by Factoring Goal Solve quadratic equations. VOCABULARY Monomial An expression that is either a number, a variable, or the product of a number and one or more variables Binomial The sum of two monomials Trinomial The sum of three monomials Quadratic equation An equation in one variable that can be written in the form ax 2 + bx + c = 0 where a 0 Root of an equation A solution of a quadratic function Zero of a function The numbers p and q of a function in intercept form are also called the zeros of the function. FACTOR 2 x bx c STEP 1: STEP 2: c Example 1 b Find two numbers whose product is c r r c 1 2 1 2 and whose sum is b. r r b r c r 1 2 b Write in factored form. ( x r )( x r ) 1 2
Factor trinomials of the form x 2 bx c Factor the expression x 2 7x 8. SPECIAL FACTORING PATTERNS Pattern Name Difference of Two Squares Perfect Square Trinomial Perfect Square Trinomial a 2 b 2 = ( a + b )( a b ) x 2 4 = (x + 2)(x 2) a 2 + 2ab + b 2 = ( a + b ) 2 x 2 + 6x + 9 = (x + 3) 2 a 2 2ab + b 2 = ( a b ) 2 x 2 4x + 4 = (x 2) 2 Example 2 Factor with special patterns Checkpoint Factor the expression. If it cannot be factored, say so. 1. x 2 + 7x + 12 2. x 2 81
ZERO PRODUCT PROPERTY Words If the _product_ of two expressions is zero, then _one_ or _both_ of the expressions equals zero. Algebra If A and B are expressions and AB = _0_, then A = _0_ or B = _0_. Example If (x + 5)(x + 2) = 0, then x + 5 = 0 or x + 2 = 0. That is, x = _ 5_ or x = _ 2_. Example 3 Find the roots of an equation Find the roots of the equation x 2 2x 15 = 0. Example 4 Find the zeros of a quadratic function Find the zeros of the function y = x 2 5x 6 by rewriting the function in intercept form. Checkpoint Complete the following exercises. 3. Find the roots of the equation x 2 3x + 2 = 0. 4. Find the zeros of the function y = x 2 + 3x 40 by rewriting the function in intercept form.
4.4 Solve ax 2 + bx + c = 0 by Factoring Goal Use factoring to solve equations of the form ax 2 + bx + c = 0. Step1: Step 2: ac Find two numbers whose product is c r r c 1 2 and whose sum is b. r r b 1 2 b r ac r 1 2 a a b Write in factored form. ( ax r )( ax r ) 1 2 Example 1 Factor ax 2 + bx + c where c > 0 Factor 2x 2 + 9x + 7. Example 2 Factor ax 2 + bx + c where c < 0 Factor 3x 2 x 2.
Checkpoint Factor the expression. If it cannot be factored, say so. 1. 3x 2 + 7x 20 2. 5x 2 13x + 6 Example 3 Factor with special patterns Factor the expression. a. 16x 2 36 b. 9y 2 + 42y + 49 c. 25t 2 110t + 121 Checkpoint Factor the expression. 3. 16y 2 40y + 25 4. 4x 2 81
Example 4 Factor out monomials first Factor the expression. a. 4x 2 4 b. 3y 2 18y c. 4m 2 10m + 24 d. 5z 2 25z + 40 Checkpoint Factor the expression. 5. 36x 2 16 6. 15p 2 + 24p 63 Example 5 Solve quadratic equations a. 2x 2 x 21 = 0
b. 4r 2 18r + 24 = 6r 12 Checkpoint Solve the equation. 7. 2x 2 + 4x 30 = 0 8. z 2 + 13z + 12 = 5z 4
4.5 Solve Quadratic Equations by Finding Square Roots Goal Solve quadratic equations by finding square roots. VOCABULARY Square root A number r is a square root of a number s if r 2 = s. Radical An expression of the form s where s is a number or expression Radicand The number s beneath the radical sign Rationalizing the denominator The process of eliminating a radical from the denominator of a fraction Conjugates The expressions a + b and a b, used to rationalize the denominator, whose product is always a rational number PROPERTIES OF SQUARE ROOTS (a > 0, b> 0) Product Property Example ab = a b 18 9 2 3 2 Quotient Property a a 2 2 = 2 b 25 25 5 b Example 1 Use properties of square roots a. 24 5 18 b. 9 64
Your Notes Example 2 Rationalize denominators of fractions Simplify (a) 7 3 and (b). 4 5 3. Checkpoint Simplify the expression. 1. 5 10 2. 9 11 Example 3 Solve a quadratic equation 1 Solve (y 6) 2 = 8. 4
Example 4 Model a dropped object with a quadratic function Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk? h = 16t 2 + h 0 Checkpoint Complete the following exercises. 3. Solve the equation 2x 2 16 = 34. 4. In Example 4, suppose that the water balloon is dropped from a height of 27 feet. How long does it take for the balloon to hit the sidewalk?
4.6 Perform Operations with Complex Numbers Goal Perform operations with complex numbers. VOCABULARY Imaginary unit i The imaginary unit i is defined as i 1. Complex number A number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part. Imaginary number A complex number a + bi where b 0 Complex conjugates Two complex numbers of the form a + bi and a bi Complex plane A coordinate plane where each point (a, b) represents a complex number a + bi. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Absolute value of a complex number The absolute value of a complex number z = a + bi, denoted z, is a nonnegative real number defined as z 2 b 2 a. THE SQUARE ROOT OF A NEGATIVE NUMBER Property 1. If r is a positive real r number, i r. then Example 3 i 3 2. By Property (1), it follows that r 2 2 i r. i 3 2 i 3
Example 1 Solve a quadratic equation 2x 2 + 15 = 35 Checkpoint Solve the equation. 1. 3x 2 + 13 = 23 SUMS AND DIFFERENCES OF COMPLEX NUMBERS To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately. Sum of complex numbers: (a + bi) + (c + di) = (a + c) + (b + d)i Difference of complex numbers: (a + bi) (c + di) = (a c) + (b d)i
Example 2 Add and subtract complex numbers Write as a complex number in standard form. a. (6 + 3i) (4 /) b. (2 + 5i) + (7 2i) Example 3 Multiply complex numbers Write the expression (2 + i)( 5 + 2i) as a complex number in standard form.. Example 4 Divide complex numbers Write the quotient 6 4 i in standard form. 2 i Checkpoint Write the expression as a complex number in standard form. 2. (12 2i) (16 + 3i) 3. 4i(9 + 5i)
4. 8 4 i 3 i 5. (4+4i) + ( 6+3i) Example 5 Plot complex numbers Plot the complex numbers in the same complex plane. a. 4 + 3i b. 5 4i ABSOLUTE VALUE OF A COMPLEX NUMBER The absolute value of a complex number z = a + bi, denoted z, is a _nonnegative_ real number defined as z = 2 2 a b. This is the distance of z from the _origin in the complex plane. Example 6 Find absolute values of complex numbers Find the absolute value of (a) 6 8i and (b) 6i.
Checkpoint Plot the complex numbers in the same complex plane. Then find the absolute value. 6. 4i 7. 2+3i 8. 3+i
4.7 Complete the Square Goal Solve quadratic equations by completing the square. VOCABULARY Completing the square The process that allows you to write an expression of the form x 2 + bx as the square of a binomial COMPLETING THE SQUARE Words To complete the square for the expression x 2 + bx, add b. 2 Algebra x 2 b + bx + = 2 2 b b x x 2 2 = 2 x b 2 2 Example 1 Solving equations by square rooting. 2 x 6x 9 1 Checkpoint Solve the equation by finding square roots. 2 1. x 10x 25 4
Example 2 Make a perfect square trinomial Find the value of c that makes x 2 12x c a perfect square trinomial. Then write the expression as the square of a binomial. Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 2. x 2 24x + c 3. x 2 + 10x +c
Example 3 Solve ax 2 bx c = 0 when a = 1 Solve x 2 10x 13 0 by completing the square. Checkpoint Solve the equations by completing the square. 4. x 2 8x + 7 = 0 Example 4 Write a quadratic function in vertex form Write y = x 2 14x 44 in vertex form. Then identify the vertex.
Checkpoint Write the quadratic function in vertex form. Then identify the vertex. 5. y = x 2 12x + 22 6. y = x 2 + 16x +53 Example 5 (Day 2) Solve ax 2 bx c 0 when a 1 Solve 3x 2 12x 27 = 0 by completing the square. Checkpoint Solve the equations by completing the square. 5. 2x 2 20x + 24 = 0
4.8 Use the Quadratic Formula and the Discriminant Goal Solve quadratic equations using the quadratic formula. VOCABULARY Quadratic formula The formula that gives the solutions to any quadratic equation Discriminant The expression b 2 4ac under the radical sign of the quadratic formula THE QUADRATIC FORMULA Let a, b, and c be real numbers such that a 0. The solutions of the quadratic equation ax 2 + bx + c are: x = b 2 b 4 2 a ac Example 1 Solve an equation with two real solutions Solve x 2 7x 6. Example 2 Solve an equation with one real solution Solve 2x 2 8x 8 = 0.
Checkpoint Use the quadratic formula to solve the equation. 1. 2x 2 + 12x = 16 2. 4x 2 13x = 7x 25 3. 3x 2 6x + 6 = 0 4. x 2 3x + 3 = 0 USING THE DISCRIMINANT OF ax 2 + bx + c = 0 When b 2 4ac > 0, the equation has _two real solutions_. The graph has _two_x-intercepts. When b 2 4ac = 0, the equation has _one real solution_. The graph has _one_x-intercept. When b 2 4ac < 0, the equation has _two imaginary solutions_. The graph has _no_x-intercepts.
Example 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x 2 + 6x + 5 = 0 b. x 2 + 6x + 9 = 0 c. x 2 + 6x + 13 = 0 Checkpoint Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 5. x 2 8x + 17 = 0 6. x 2 + 4x + 3 = 0 7. x 2 + 2x 1 = 0 8. x 2 + 6x + 4 = 0