How do I find a quadratic equation given 2 points and no vertex?
While it is true that three equations are needed to find the three coefficients, some conditions might help develop a specific equation. However, you are asking “a” quadratic equation.Assuming a vertical axis of symmetry, the equation would be of the form y = ax^2 + bx + c which can also be written as y = a(x - h)^2 + k.Case 1: If the two points have the same y value,then the axis of symmetry will pass through the midpoint of the segment between them. For example, if the given points are (1, 7) and (5, 7), then the midpoint is found by averaging the x coordinates. (1 + 5)/2 = 3. The axis of symmetry is x = 3.Two equations can then be written from the given points(1, 7) 7 = a(1 - 3)^2 + k or 7 = 4a + k(5, 7) 7 = a(5 - 3)^2 + k or 7 = 4a + kAs expected, we have only one equation with two unknowns, resulting in no particular solution. However, once again, looking for a solution, choose a value for k, say -5, the equation becomes 7 = 4a - 512 = 4aa = 3The equation becomes y = 3(x - 3)^2 -5which becomes y = 9(x^2 -6x +9) -5and then y = 9x^2 -54x + 76.Case 2: If the two points have the same y value,Example (1, 7) and (3, 31)7 = a(1)^2 + b(1) + c and 31 = a(3)^2 + b(3) +c(1) 7 = a + b + c and (2) 31 = 9a + 3b + cSubtracting equation 1 from equation 224 = 8a + 2bdividing by 212 = 4a + bb = 12 - 4aLet a = 2 or any other chosen valueb = 12 - 4(2)b = 4Substituting into equation 17 = (2) + (4) + cc = -1Answer: An equation going through the two points is y = 2x^2 + 4x - 1
How do I determine the equation of the tangent to the curve [math] y=2x-x^2 [/math] that passes through point [math] (2,9) [/math]?
There are actually two such tangents:Let y = mx + n be the tangent we're looking for, i.e. we have to determine m and n.By definition of tangent, the line y = mx + n is a tangent to the given curve if they interesect in exactly one point. So let's equate the two equations to calculate intersection(s):[math]y = mx + n = 2x - x^2,[/math][math]x^2 + (m-2)x + n = 0,[/math][math]x = \frac{m-2}{2}\pm\sqrt{\left(\frac{m-2}{2}\right)^2-n}.[/math]If the radicand (the "thing" under the square root) is positive, there will be two intersections. If it is negative, there will be no intersection. If it is zero, there will be one intersection, and the line will indeed be a tangent.So we get:[math]\left(\frac{m-2}{2}\right)^2-n=0,[/math][math]n=\left(\frac{m-2}{2}\right)^2.[/math]Now we can use the fact that the tangent passes through (2,9):[math]9=2m+n=2m+\left(\frac{m-2}{2}\right)^2=2m+\frac{\left(m-2\right)^2}{4},[/math][math]36=8m+4-4m+m^2,[/math][math]m^2+4m-32=0,[/math][math]m=-2\pm\sqrt{4+32}=-2\pm\sqrt{36}=-2\pm 6,[/math][math]m=4\ \text{or}\ m=-8.[/math]For m = 4, we get[math]n=\frac{(2-4)^2}{4}=\frac{4}{4}=1,[/math]and for m = -8, we get[math]n=\frac{(2-(-8))^2}{4}=\frac{100}{4}=25.[/math]The two solutions are therefore y = 4x + 1 and y = -8x + 25.
A line makes an angle of 45 degrees with the x-axis. What is the slope of the line?
hey bro,I have assume that you don't know the meaning of slope, so let us find outslope: it is the change of something ( say y), with respect to some other thing ( say x). example: velocity is the rate of change of displacement with respect to time.in the same way if we take this knowledge and relate it with the graphical representation.we will get to know that the line ( say y) when is elevated from the origin to 45 degrees with respect to x axis, then slope is equal to the change in y divided by change in x. By using trigonometric ratios tanθ = change in y divided by change in x.hence we can conclude that tanθ = slope.in your case θ = 45 degrees.therefore, slope = tan45and slope = 1 ……………………………………………(tan45=1)hope you are totally satisfied with this contempt answer.this once is not for the person who asked question.the writer takes a deep breathe in and …….. submits the answer.
A rectangular sheet of paper of dimensions 25cm by 14cm is curled along its width. What is the volume of the cylinder thus formed?
The volume of a cylinder is equal to the area of the circular base times the height:[math]V=(A)(H)[/math]or[math]V=(\pi R^2)(H)[/math] ————- equation 1The height of the cylinder will be the length of the long side of your paper, so [math]H=25cm[/math]. Now we need the radius of the circular base.The width of the paper will equal the circumference of the circle which forms the base of the cylinder. Use this to determine the radius of the circle:Circumference = [math]2 \pi R[/math][math]\therefore[/math] [math]14cm=2 \pi R[/math]or[math]R=\frac{14}{2 \pi } =\frac{7}{\pi}[/math] cm.Plug this into equation 1 (above):[math]V=\pi (\frac{7}{\pi })^2 (25)=\frac{(49)(25)}{\pi}=389.9[/math][math] cm^3[/math]
What are the officers of the Roman army called?
From the site listed below... "Let's start from the bottom. The lowest role of leadership on the chain is that of Decuriones. This man was the senior officer in charge of ten horsemen. He would basically represent that small group of soldiers and, if he was killed in battle, the next most senior person would take his place. The next step up on the ladder is the job of Praefectus. This person was the commander of the cavalry. One more rung on the ladder is the Centurion. He would lead a century. This officer was non-commisioned, and to hold this title, would have to had previously been a sergeant. The following are even higher positions that are reached through an entirely different type of promotion. Only senators could ever get these positions. The Military Tribune is the first on this list. The Tribune was in charge of keeping an accurate list of all the soldiers, noting casualties, and when needed becoming a mediator in minor disputes. The Qaestor was the treasurer of the Roman Army, they would handle the logistics. These people served a term of one year. The commander and chief of a single roman legion, also known as the legion general. The second highest in command would be the Legati. This person served as the lieutenant general. This person was usually between the age of 30 and 40, and would serve a term of 3-4 years. The head man in charge of the ENTIRE Roman Army was the Dux. This person had it all. The Dux has the power to crush civilizations. The Dux has the power to expand the great Empire of Ancient Rome." Aloha