A parabola has a vertex at #(5, 4)# and passes through #(6, 13/4)#. What are the x-intercepts?
1 Answer
Mar 2, 2017
The x-intercepts are given by
Explanation:
Start by finding the equation of the parabola. The vertex form of a parabola, with vertex
#y = a(x- p)^2 + q#
We know an x-value, a y-value and the vertex. We can therefore set up an equation and solve for
#13/4 = a(6 - 5)^2 + 4#
#13/4 = a(1)^2 + 4#
#-3/4 = a#
The equation is therefore
#y = -3/4(x - 5)^2 + 4#
We can solve for the x-intercepts by taking the square root. Set
#0 = -3/4(x - 5)^2 + 4#
#-4 = -3/4(x - 5)^2#
#-4/(-3/4) = (x - 5)^2#
#16/3 = (x - 5)^2#
#+-4/sqrt(3) = x - 5#
#x = 5 +- 4/sqrt(3)#
#x = 5 +- (4sqrt(3))/3#
A graphical depiction of the parabola confirms our findings.
graph{y = -3/4(x - 5)^2 + 4 [-10, 10, -5, 5]}
Hopefully this helps!