Section 2.2 Flashcards
What is the equation to find the vertex of a function in the form ax2+bx+c?
The equation is -b/2a for the x coordinate and f of -b/2a for the y coordinate. (f of -b/2a means that you plug in the vertex x value into the function and solve for y. y is the y coordinate of the vertex.
The domain of any quadratic equation include what numbers?
All real numbers.
How do you express all real numbers in interval notation?
(-infinity, infinity)
The minimum value is what value of the function.
The lowest value.
The range is greater than or equal to what value?
The minimum value.
How do we find a pair of numbers whose sum is 10 and whose product is as large as possible?
First set up two equations, xy=p and x+y=10 then solve for y in x + y =10. Then subtitute y in y in xy=p which equals x(10-x). Then distribute the function giving you -x2+10x. Then find the x coordinate of the vertex using -b/2a. Then plug your x value into your y=10-x equation to find your second value. The pair is 5, and 5 whose product equals 25.
Among all pairs of numbers whose difference is 22, find a pair whose product is as small as possible?
First set up two equations, xy=p and x-y=22 then solve for y in x -y =22. Then subtitute y in y in xy=p which equals x(x-22). Then distribute the function giving you x2-22x. Then find the x coordinate of the vertex using -b/2a. Then plug your x value into your y=x-22 equation to find your second value. The pair is 11, and -11 whose product equals -121.
Farmer Ed has 700 meters of fencing, and wants to enclose a rectangular plot that borders on a river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?
Your length = 700-x. Your width = x. Multiply x(700-x) to get the area function. Find the x value of the vertex using -b/2a to find the maximum x value (width) of the area function which is 175. To find your length plug in your x value (175) into your length equation (700-x) which gives you 350. To get your maximum area find your y value (the maximum) of the area function’s vertex. To get this you plug in x into the quadratic equation and solve for y which is 61,250.
Peter has 80 yards of fencing to enclose a rectangular area. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
Area equals x times y and perimeter equals 2x plus 2y. Then you use the set the perimeter to 80 and solve for y which gives you y=40-x. Then you use the area equation and set your y = to 40 -x giving you a=x(40-x). Then you distribute and set the equation to general form. Then you solve for x using the vertex formula -b/2a which gives you 20. Then you plug this value into the width dimension equation 40-x which gives you 20. Your length is 40 and you width is 20 and your area is xy which is 400.
A rectangular playground is to be fenced off and divided in two by another fence parrallel to one side of the playground. 240 ft of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?
You know that area is equal to xy. You set the perimeter equation as 3x + 2y = 240 and solve for y which =120-3/2x. You plug y into the area equation, distribute, and set the quadratic equation to general form. You then find the maximum length by solving for x by using the vertex formula -b/2a. You then plug x into your width formula and solving for y which gives you 60. You then use the area formula to get the area which is 2400.
