Writing Quadratic Functions in Standard Form

Chapter Summar Ke Terms standard form (general form) of a quadratic function (.1) parabola (.1) leading coefficient (.) second differences (.) vertical motion model (.3) zeros (.3) interval (.3) open interval (.3) closed interval (.3) half-closed interval (.3) half-open interval (.3) factor an epression (.) factored form (.) verte (.5) ais of smmetr (.5) verte form (.6) vertical dilation (.7) dilation factor (.7).1 Writing Quadratic Functions in Standard Form A quadratic function written in standard form, or general form, is in the form f() 5 a 1 b 1 c, where a fi. In this form, a and b represent numerical coefficients and c represents a constant. Cara wants to make an herb garden with a rectangular area in the center and equal square sections on both sides. The entire length of the garden can be no longer than feet. Let g represent the width of each square section. The length would be g. The area of the rectangular part of the garden will be g( g). Use the Distributive Propert to rewrite the epression: g g. Write the area as a function in standard form: A(g) 5 g 1 g 1. a 5 ; b 5 ; c 5 97

.1 Identifing Maimums and Minimums of Quadratic Graphs The shape that a quadratic function forms when graphed is called a parabola. A parabola is a smooth curve in a U-shape that is upside-down when a is negative and right-side-up when a is positive. You can use a graphing calculator to identif the absolute maimum or minimum of a quadratic function. The function A(g) 5 g 1 g has an absolute maimum of (5, 5). (5, 5) 3 1 16 1 16 3. Creating and Analzing Linear and Quadratic Graphs You have feet of fencing with which to enclose an area. The table of possible lengths and widths represents a linear function. The table with possible lengths and areas represents a quadratic function. Length (ft) Width (ft) 1 1 9 3 7 6 5 5 6 7 3 9 1 1 Width (ft) 1 16 1 1 6 5 1 6 1 1 16 Length (ft) 1 The -intercept of 1 indicates the width when the length is feet. The -intercept of 1 indicates the length when the width is feet. Neither point is appropriate for an enclosed area. 9 Chapter Introduction to Quadratic Functions

Length (ft) Area (ft ) 1 9 16 3 1 5 5 6 7 1 16 9 9 1 Area (sq ft) 5 35 3 5 15 1 5 5 1 1 6 1 1 16 Length (ft) 1 The -intercepts are and 1, indicating the length when the area is feet. Neither point is appropriate for an enclosed area. The -intercept is, indicating the area when the length is.. Identifing Linear and Quadratic Functions First differences are the differences between successive output values when successive input values have a difference of 1. Second differences are the differences between consecutive values of first differences. Linear functions have constant first differences and second differences of. Quadratic functions have changing first differences and constant second differences. Linear function: 5 1 Quadratic function: 5 3 1 1 3 1 First Differences 1 1 1 1 Second Differences 1 1 3 9 First Differences 1 3 7 11 Second Differences Chapter Summar 99

.3 Identifing and Describing the Domain and Range of a Quadratic Function The domain of a function is all the possible -values. The range of a function is all the possible -values. The domain and range can be different for the same function when considering the mathematical function alone compared to considering the contetual situation. A tennis ball is hit into the air from 3 feet above ground with a vertical velocit of 65 feet per second. The function that describes the height of the ball in terms of time is g(t) 5 16t 1 65t 1 3. 6 Height (ft) 6 Time (s) The domain is all real numbers from negative infinit to positive infinit. In terms of the problem situation, the domain is all real numbers greater than or equal to and less than or equal to about.1. The range is all real numbers less than or equal to about 69. In terms of the problem situation, the range is all real numbers less than or equal to about 69 and greater than or equal to. 93 Chapter Introduction to Quadratic Functions

.3 Identifing the Zeros of a Quadratic Function and the Roots of a Quadratic Equation The -intercepts of a graph of a quadratic function are also called the zeros of the quadratic function. The points where the graph crosses the -ais are called the -intercepts. A firework is launched into the air from the ground with a vertical velocit of feet per second. The function that describes the height of the firework in terms of time is g(t) 5 16t 1 t. The -intercepts, or zeros are (, ) and (, ). 5 35 3 5 15 1 5 1 3 5 6 7 9 Chapter Summar 931

.3 Determining Intervals of Increase and Decrease of a Quadratic Function The absolute maimum or absolute minimum is the turning point of a parabola. If the quadratic function has an absolute maimum, the -value increases to the maimum and then decreases. If it has an absolute minimum, the -value decreases to the minimum and then increases. You can use interval notation to describe the interval of the domain in which the function is increasing and the interval of the domain in which the function is decreasing. An interval is defined as the set of real numbers between two given numbers. An open interval (a, b) describes the set of all numbers between a and b, but not including a or b. A closed interval [a, b] describes the set of all numbers between a and b, including a and b. A half-closed or half-open interval (a, b] describes the set of all numbers between a and b, including b but not including a. Or, [a, b) describes the set of all numbers between a and b, including a but not including b. Intervals that are unbounded are written using the smbol for infinit, `. 16 6 6 16 The graph represents the function f() 5. Domain: All real numbers Range: All real numbers greater than or equal to.5 -intercept: (, ) Zeros: (3, ), (, ) Interval of decrease: ( `, 1 ) Interval of increase: ( 1, ` ) 93 Chapter Introduction to Quadratic Functions

. Factoring the Greatest Common Factor from an Algebraic Epression To factor an epression means to use the Distributive Propert in reverse to rewrite the epression as a product of factors. When factoring algebraic epressions, ou can factor out the greatest common factor from all the terms. When factoring an epression that contains a negative leading coefficient, it is convention to factor out the negative sign. 5 1 5 5 5(5) 1 5(17) 3 1 7 5 3() 1 (3)(9) 5 5(5 1 17) 5 3( 9). Writing Quadratic Functions in Factored Form A quadratic function written in factored form is in the form f() 5 a( r 1 )( r ), where a fi. f() 5 1 9 1 1 5 1( 1 )( 1 7). Writing a Quadratic Function in Factored Form Given Its -Intercepts A quadratic function with -intercepts (r 1, ) and (r, ), can be written in factored form as f() 5 a( r 1 )( r ). The sign of a tells ou whether the parabola opens upward (positive a) or downward (negative a). The quadratic function in factored form of a parabola that opens downward and has zeros at (3, ) and (, ) is: f() 5 a( 3)( 1 ) for a,. Determining -Intercepts from Functions in Factored Form For a quadratic function written in factored form f() 5 a( r 1 )( r ), the variables r 1 and r represent the -coordinates of the -intercepts. The -intercepts are (r 1, ) and (r, ). g() 5 ( )( 1 3) The function in factored form is g() 5 ( )( 1 3). The zeros are (, ) and (3, ). Chapter Summar 933

.5 Determining the Ais of Smmetr of Quadratic Functions The ais of smmetr of a parabola is the vertical line that passes through the verte and divides the parabola into two mirror images. The equation of that line is 5 the -coordinate of the verte. The value of the ais of smmetr is the average of the two -intercepts. The -intercepts of a parabola are (3, ) and (7, ). The ais of smmetr is 5 because 3 1 7 5 5..5 Determining the Verte of Quadratic Functions The verte of a parabola is the lowest or highest point on the curve. The equation of the ais of smmetr is 5 the -coordinate of the verte. The -coordinate of the verte can be found b substituting the -value into the equation and solving for. The equation of a parabola is f() 5 6 1 7. The ais of smmetr is 5 3, so the -coordinate of the verte is 3. The -coordinate when 5 3 is: f(3) 5 (3) 6(3) 1 7 5 9 1 1 1 7 5 16 The verte is at (3, 16)..5 Determining the Ais of Smmetr Using Smmetric Points The -coordinate of the verte is halfwa between the -coordinates of smmetric points on a parabola. In other words, the -coordinate of the verte is the average, or midpoint, of the -coordinates of points on the parabola with the same -coordinates. The equation of a parabola is f() 5 1 3 and two smmetric points on the parabola are (7, ) and (, ). The ais of smmetr is 5 3 because 7 1 5 3. 93 Chapter Introduction to Quadratic Functions

.5 Determining Smmetric Points on the Parabola Using a Verte The -coordinates of smmetric points on a parabola are the same. Smmetric points are the same distance from the ais of smmetr. So, the -coordinate of the verte is the average, or midpoint, of the -coordinates of points on the parabola with the same -coordinates. The verte of a parabola is (3, 5). A point on the parabola is (, 3). Another point on the parabola is (6, 3) because: 1 a 5 3 1 a 5 6 a 5 6.6 Identifing the Verte of a Quadratic Function in Verte Form A quadratic function written in verte form is in the form f() 5 a( h) 1 k, where a fi. The variable h represents the -coordinate of the verte. The variable k represents the -coordinate of the verte. f() 5 1 15 verte: (, 1) verte form: f() 5 ( ) 1 zero(s): (3, ) and (5, ) factored form: f() 5 ( 3)( 5) Chapter Summar 935

.6 Identifing Characteristics of a Parabola Given Its Equation in Different Forms In a quadratic function written in standard form, the sign of the a value tells ou whether the parabola opens up or down. The c value tells ou the -intercept. In a quadratic function written in factored form, the sign of the a value tells ou whether the parabola opens up or down. The r 1 and r values tell ou the -intercepts. In a quadratic function written in verte form, the sign of the a value tells ou whether the parabola opens up or down. The variable h represents the -coordinate of the verte and the variable k represents the -coordinate of the verte. a. f() 5 ( 1 3) 1 6 The function is in verte form. The parabola opens up and the verte is (3, 6). b. f() 5 1 5 1 1 The function is in standard form. The parabola opens down and the -intercept is (, 1). c. f() 5 ( 1 )( 5) The function is in factored form. The parabola opens up and the zeros are (, ) and (5, )..6 Writing an Equation of a Parabola Given Information about Its Graph Given information about whether the parabola opens up or down, the -intercept, -intercepts, or verte, ou can write an equation in the appropriate form. a. The zeros are 3 and 7, and the parabola opens up. f() 5 a( 3)( 7) for a. b. The verte is (, ), and the parabola opens down. f() 5 a( ) 1 for a, 936 Chapter Introduction to Quadratic Functions

.7 Translating Quadratic Functions Vertical translations are performed on a basic quadratic function g() 5. Adding to the equation translates it up and subtracting translates it down. Horizontal translations are performed on the argument,, of a basic quadratic function. g() 5 basic function c() 5 1 g() translated units up, so (, ) (, 1 ). d() 5 g() translated units down, so (, ) (, ). 6 6 6 c() 5 1 g() 5 6 d() 5 g() 5 j() 5 ( 1 ) k() 5 ( ) basic function g() translated units left, so (, ) (, ). g() translated units right, so (, ) ( 1, ). 6 6 6 j() 5 ( 1 ) k() 5 ( ) g() 5 6 Chapter Summar 937

.7 Reflecting Quadratic Functions Multipling the basic quadratic function b 1 results in a reflection over the line 5. Multipling the argument of the basic quadratic function b 1 results in a reflection over the line 5, which ends up being the same as the original function because squared values are alwas positive. g() 5 m() 5 n() 5 () basic function g() is reflected over 5, so (, ) (, ). g() is reflected over 5, so (, ) (, ). 6 6 g() 5 n() 5 () 6 m() 5 6 93 Chapter Introduction to Quadratic Functions

.7 Dilating Quadratic Functions A vertical dilation of a function is a transformation in which the -coordinate of ever point on the graph of the function is multiplied b a common factor called the dilation factor. A vertical dilation stretches or shrinks the graph of a function. When the dilation factor, a, is greater than 1, the graph of the function appears to be stretched verticall. When, a, 1, the function appears to shrink verticall. You can use the coordinate notation shown to indicate a vertical dilation. (, ) S (, a), where a is the dilation factor. g() 5 basic function v() 5 g() stretched b a dilation factor of, so (, ) S (, ). w() 5 1 g() shrunk b a dilation factor of 1, so (, ) S (, 1 ). 1 w() 5 6 6 6 v() 5 g() 5 6 Chapter Summar 939

.7 Writing Equations Given Transformations Use given characteristics including a transformation done to a basic equation to write a quadratic equation in verte form and sketch its graph. The function is quadratic. The function is continuous. The parabola opens upward. The function is translated units to the left and 1 unit up from f() 5. The function is verticall dilated with a dilation factor of 3. Equation: f() 5 3( 1 ) 1 1 16 1 1 6 6 6 9 Chapter Introduction to Quadratic Functions