A problem involving the intersection of a parabola and a line is presented along with detailed solution.
Problem:
The graphs of a parabola given by y = f(x) and that of a line given by y = g(x) are shown below. The parabola and the line intersect at points A and D; where A is on the x-axis and has an x-coordinate equal to 2. V is the vertex of the parabola.
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a) Find the equation of g and write it of the form g(x) = m x + b
b) Find the equation of the parabola.
c) Find x coordinate of point D.
d) Use the graph to solve the inequality f(x) > g(x).
Solution to Problem 1:
a) y = g(x) is the equation of a line through the points C(-3,-5) and A(2,0)
slope = (0-(-5)) / (2 -(-3)) = 1
equation of line: y - 0 = 1(x - 2) , which may be written as: y = x - 2
hence g(x) = x - 2
b) The vertex of the parabola is known V(4,-4). The equation of the parabola in vertex form is given by
y = a(x - 4)^{2} - 4
Point A(2,0) is on the parabola hence: 0 = a(0 - 4)^{2} - 4
solve the above equation for a to find a = 1
hence the equation of the parabola is given by: y = (x - 4)^{2} - 4
c) The x coordinate of point D is determined by solving the equation
(x - 4)^{2} - 4 = x - 2 (intersection of line and the parabola)
solve the above equation to obtain the solutions x = 2 and x = 7.
x = 2 is already known, it is the x coordinate of A. x = 7 is the x coordinate of point D.
d) From the left side of the graph to point A, f(x) > g(x) and therefore the interval (-infinity , 2) is a set of solutions. Also from point D to the right side of the graph f(x) > g(x) and therefore another set of solutions is given by the interval (7 , + infinity).
The solution set of the inequality f(x) > g(x) is given by: