How do you find the intercept and vertex of #f(x)= -4x^2 + 4x + 4#?

Question

How do you find the  intercept and vertex of #f(x)= -4x^2 + 4x + 4#?

Isaiah Anthony · Accepted Answer

Vertex: #(-1/2, 5)#

#y# intercept: #(0, 4)#

#x# intercepts: #(1/2+-sqrt(5)/2, 0)# Given: #f(x) = -4x^2+4x+4# In order to transform this into a vertex, we can finish the square: #f(x) = -4x^2+4x+4# #color(white)(f(x)) = -4(x^2-x-1)# #color(white)(f(x)) = -4(x^2-x+1/4-5/4)# #color(white)(f(x)) = -4((x-1/2)^2-5/4)# #color(white)(f(x)) = -4(x-1/2)^2+5# In vertex form, this is: #f(x) = a(x-h)^2+k# where #a=-4# is the multiplier (affecting the steepness and up/down orientation of the parabola) and #(h,k) = (-1/2, 5)# is the vertex. The difference of squares identity can be utilized by us. #a^2-b^2 = (a-b)(a+b)# with #a=2(x-1/2)# and #b=sqrt(5)# to get it into factored form so we can find the zeros: #f(x) = -4(x-1/2)^2+5# #color(white)(f(x)) = -((2(x-1/2))^2-(sqrt(5))^2)# #color(white)(f(x)) = -(2(x-1/2)-sqrt(5))(2(x-1/2)+sqrt(5))# #color(white)(f(x)) = -(2x-1-sqrt(5))(2x-1+sqrt(5))# The zeros are therefore: #x = 1/2(1+-sqrt(5))# So the #x# intercepts are: #(1/2+sqrt(5)/2, 0)" "# and #" "(1/2 - sqrt(5)/2, 0)# The #y# intercept is #f(0) = 0+0+4 = 4#, i.e. #(0, 4)# graph{[-4, 4, -3.12, 6.88]} for x^2 + 4x + 4.

Isaiah Anthony · Answer

Vertex #(0.5, 5)#
Intercept #(0,4)# Given - #f(x)=-4x^2+4x+4# y - intercept At x = 0; #y=-4(0)^2+4(0)+4=0# #y=4# Intercept #(0,4)# Vertex #x=(-b)/(2xxa)=(-4)/(2 xx(-4))=(-4)/(-8)=1/2=0.5# At #x=0.5# #y=-4(0.5)^2+4(0.5)+4=-1+2+4=5# Vertex #(0.5, 5)#

Isaiah Anthony · Answer

undefined To find the intercepts of the function f(x) = -4x^2 + 4x + 4, you can set y = 0 to find the x-intercepts (roots) and x = 0 to find the y-intercept. To find the vertex, you can use the formula x = -b / (2a) to find the x-coordinate of the vertex, and then substitute that x-value into the original function to find the corresponding y-coordinate.

1. X-intercepts (Roots):
   Set y = 0: -4x^2 + 4x + 4 = 0
   Solve for x using factoring, completing the square, or the quadratic formula.

2. Y-intercept:
   Set x = 0: f(0) = -4(0)^2 + 4(0) + 4 = 4
   So, the y-intercept is (0, 4).

3. Vertex:
   Use the formula x = -b / (2a) to find the x-coordinate of the vertex.
   Once you have the x-coordinate, substitute it into the original function to find the y-coordinate.