Find the zeros of f x x 2 2x 3 by using a graph and table . Method 2 Use a calculator , Enter y x 2 2x 3 into a graphing calculator . Both the table and the graph show that y 0 at x 3 and x 1 . These are the zeros of the function , 1 Find the zeros of g x x 2 2x 3 by using a graph and. You can also find zeros by using algebra For example to find the zeros of. Functions have zeros f x x 2 2x 3 you can set the function equal to zero The solutions to. or x intercepts the related equation x 2 2x 3 0 represent the zeros of the function . Equations have, The solutions to a quadratic equation of the form ax 2 bx c 0 are roots The. solutions or roots , roots of an equation are the values of the variable that make the equation true . You can find the roots of some quadratic equations by factoring and applying the. Zero Product Property , Zero Product Property, For all real numbers a and b . WORDS NUMBERS ALGEBRA, If the product of two quantities 3 0 0 If ab 0 then a 0. equals zero at least one of the 0 4 0 or b 0 , quantities equals zero . EXAMPLE 2 Finding Zeros by Factoring, Find the zeros of each function by factoring . A f x x 2 8x 12, x 2 8x 12 0 Set the function equal to 0 . x 2 x 6 0 Factor Find factors of 12 that add to 8 . x 2 0 or x 6 0 Apply the Zero Product Property , x 2 or x 6 Solve each equation . x 2 8x 12 0, x 2, 8x 12 0, , Substitute each, 2 2 8 2 12 0 6 8 6 12 0. 2 value into, 4 16 12 0 36 48 12 0 the original, 0 0 0 0 . 334 Chapter 5 Quadratic Functions, a207se c05l03 0333 0340 indd 334 12 14 05 1 50 10 PM. Find the zeros of each function by factoring , B g x 3x 2 12x. 3x 2 12x 0 Set the function equal to 0 , 3x x 4 0 Factor The GCF is 3x . 3x 0 or x 4 0 Apply the Zero Product Property , x 0 or x 4 Solve each equation . Check Check algebraically and by graphing , 3x 2 12x 0. 3x 2 12x 0, x x , 3 0 12 0 0, 3 4 12 4 0, 0 0 0 48 48 0 . Find the zeros of each function by factoring , 2a f x x 2 5x 6 2b g x x 2 8x. Any object that is thrown or launched into the air such as a baseball basketball . or soccer ball is a projectile The general function that approximates the height h. in feet of a projectile on Earth after t seconds is given below . Constant due to Earth s Initial vertical velocity Initial height. gravity in ft s2 in ft s at t 0 in ft at t 0 , Note that this model has limitations because it does not account for air. resistance wind and other real world factors , EXAMPLE 3 Sports Application. A soccer ball is kicked from ground level with an initial vertical velocity of. 32 ft s After how many seconds will the ball hit the ground . h t 16t 2 v 0t h 0 Write the general projectile function . h t 16t 32t 0, Substitute 32 for v 0 and 0 for h 0 . The ball will hit the ground when its height is zero . 16t 2 32t 0 Set h t equal to 0 , 16t t 2 0 Factor The GCF is 16t . 16t 0 or t 2 0 Apply the Zero Product Property , t 0 or t 2 Solve each equation . The ball will hit the ground in 2 seconds , Notice that the height is also zero when. t 0 the instant that the ball is kicked , Check The graph of the function. h t 16t 2 32t shows its , zeros at 0 and 2 , 5 3 Solving Quadratic Equations by Graphing and Factoring 335. a207se c05l03 0333 0340 indd 335 12 14 05 1 50 13 PM. 3 A football is kicked from ground level with an initial vertical. velocity of 48 ft s How long is the ball in the air . Quadratic expressions can have one two or three terms such as 16t 2 . 16t 2 25t or 16t 2 25t 6 Quadratic expressions with two terms are. binomials Quadratic expressions with three terms are trinomials Some. quadratic expressions with perfect squares have special factoring rules . Special Products and Factors, Difference of Two Squares Perfect Square Trinomial. a 2 2ab b 2 a b , a 2 b 2 a b a b , a 2 2ab b 2 a b . EXAMPLE 4 Finding Roots by Using Special Factors, Find the roots of each equation by factoring . A 9x 2 1, 9x 2 1 0 Rewrite in standard form , 3x 1 0. 2 2, Write the left side as a 2 b 2 , 3x 1 3x 1 0 Factor the difference of squares . 3x 1 0 or 3x 1 0 Apply the Zero Product Property . 1 or x , x 1 Solve each equation , 3 3, Check Graph the related function. f x 9x 2 1 on a graphing, calculator The function appears to x x. have zeros at 13 and 13 , B 40x 8x 2 50, A quadratic equation 8x 2 40x 50 0 Rewrite in standard form . can have two roots, that are equal such 2 4x 20x 25 0. Factor The GCF is 2 , as x 52 and x 52 4x 2 20x 25 0 Divide both sides by 2 . Two equal roots are, sometimes called a 2x 2 2x 5 5 0. 2 2, Write the left side as a 2 2ab b 2 , double root 2x 5 2 0 Factor the perfect square trinomial a b . 2x 5 0 or 2x 5 0 Apply the Zero Product Property . x or x 5, Solve each equation , 2 2, 5 into the original equation . Check Substitute the root , 40x 8x 50, , 100 100 . Find the roots of each equation by factoring , 4a x 2 4x 4 4b 25x 2 9. 336 Chapter 5 Quadratic Functions, a207se c05l03 0333 0340 indd 336 12 14 05 1 50 15 PM. If you know the zeros of a function you can work backward to write a rule. for the function , EXAMPLE 5 Using Zeros to Write Function Rules. Write a quadratic function in standard form with zeros 2 and 1 . x 2 or x 1 Write the zeros as solutions for two equations . x 2 0 or x 1 0 Rewrite each equation so that it equals 0 . x 2 x 1 0 Apply the converse of the Zero Product, Property to write a product that equals 0 . x2 x 2 0 Multiply the binomials , f x x x 2, Replace 0 with f x . Check Graph the function, f x x 2 x 2 on a, , calculator The graph shows. the original zeros of 2 and 1 , 5 Write a quadratic function in standard form with. zeros 5 and 5 , Note that there are many quadratic functions x. with the same zeros For example the functions, f x x 2 x 2 g x x 2 x 2 and. h x 2x 2 2x 4 all have zeros at 2 and 1 , THINK AND DISCUSS. 1 Describe the zeros of a function whose terms form a perfect square. trinomial , 2 Compare the x and y intercepts of a quadratic function with those of a. linear function , 3 Explain how a table can be used to find zeros of a function . 4 GET ORGANIZED Copy and complete the graphic organizer In each. box give information about special products and factors . i i i , vvi i Vi v , i , i viV i , , 5 3 Solving Quadratic Equations by Graphing and Factoring 337. a207se c05l03 0333 0340 indd 337 12 14 05 1 50 17 PM.

Solving Quadratic Equations by Graphing A quadratic equation is a nonlinear equation that can be written in the standard form ax2 + bx + c = 0, where a ? 0. In Chapter 7, you solved quadratic equations by factoring. You can also solve quadratic equations by graphing. Solving a Quadratic Equation: Two Real Solutions Solve x2 + 2x = 3 by ...

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MSC 47D09 DOI: 10.14529/mmp190411 NUMERICAL RESEARCH OF THE MATHEMATICAL MODEL FOR TRAFFIC FLOW A.S. Konkina, South Ural State University, Chelyabinsk, Russian Federation,

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A transistor will act linearly if the base-emitter current is within a specified range (depending on the model of transistor). That means that the collector-emitter current will be some multiple of the base-emitter current. That multiplier is called the transistor's gain. For example, the gain of a 2N3904 (a very common small transistor) is ...

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