Parabola - practice problems
A parabola is a U-shaped curve that is the graph of a quadratic function and a conic section formed by cutting a cone parallel to its side. It is defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The standard form of a parabola is y = ax² + bx + c or in vertex form y = a(x-h)² + k. The vertex represents the minimum or maximum point, while the axis of symmetry divides the parabola into mirror images. Parabolas model projectile motion, satellite dish shapes, suspension bridge cables, and headlight reflectors. Understanding parabolas is essential for optimization problems and analyzing quadratic relationships.Instructions: Solve each problem carefully and provide a detailed solution for every item.
Number of problems found: 22
- EQ2 - roots
Find the equation whose roots are the two numbers 8 and 10? - Suppose 10 Suppose 4+7i is a solution of 5z²+Az+B=0, where A, B∈R. Find A and B.
- Parabola with abs
A). Sketch the graph of the function f(x)=x * absolute(x) = x * |x| b). For what values of x is f(x) differentiable c). Find F(x) - Trapezoid and bases
Find the area of the trapezoid whose bases differ by three and whose height is three less than twice the upper base. - Parabola 3
Find the equation of a parabola with its focus at (0,2) and its vertex at the origin. f: y=x²+bx+c - EQ2 - symetric results Given the function y = x² - 4x + 3. Determine all real numbers z such that g(x) = g(-2).
- Arithmetic mean - parabola
Find the value of k so that k² + 2k – 3 is the arithmetic mean between k² + 4k + 5 and k² – 6k + 10. - Car kinetic energy
Determine the kinetic energy of a car with a mass of 800 kg if it travels at a speed: a) 10 m/s b) 20 m/s - The sequence Find the nth term of the progression 2,6,12,20...
- Accelerated motion
The position of a mass point that moves along the x-axis is given by the relation x=10t²-5t. Express its velocity and acceleration. - A missile
A missile is fired with a speed of 100 fps in a direction 30° above the horizontal. Determine the maximum height to which it rises. Fps foot per second. - Ball
The soldier fired the Ball at an angle of 57° at an initial velocity of 186 m/s. Determine the length of the litter. (g = 9.81 m/s²). - An archer
An archer stands 60 meters (m) from a target. She launches an arrow that lands 3 centimeters (cm) from the bull's eye. The archer changes her position to 40 m from the target, and her next arrow lands 2 cm from the bull's eye. She changes her position to - Ballistic curve
The soldier fired the ballistic grenade at a 45° angle. The first half ascended, and the second fell. How far and height did it reach if his average speed was 1200 km/h, and what 12 seconds were taken from the shot to impact? - Tennis ball parabola
In a tennis match, Adrien is 5 m from the net when he hits a ball 80 cm off the ground. The maximum height of its parabolic path passing through the net was 1.5 m. If the length of the court is 23.77 m, will the ball land inside the court? - Parabola focus equation
Determine the equation of the parabola that has the point F = [3,2] as its focus and the line x+y+1=0 as its shift line. - Function graph intersection Determine the value of the number a so that the graphs of the functions f: y = x² and g: y = 2x + a have exactly one point in common.
- Quadratic function graph
A quadratic function is given: y = -x² + 2x + 3 a) determine the intersections with the x, y-axis and peak V b) draw a graph and describe c) for which x applies f (x) = 3 - Parabola
Find the equation of a parabola that contains the points at A[10; -5], B[18; -7], C[20; 0]. (use y = ax²+bx+c) - Parabola point equation Write the equation of the parabola that passes through the points: A[1,1] B[3,-1] C[1,2]
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