How do you find the vertex and intercepts for #y=2(x+2)^2+3#?

Answer 1

Vertex is at #(-2,3)# , y-intercept is at #0,11# , No x-intercept.

#y=2(x+2)^2 +3 #. Comparing with standard vertex form of equation #y=a(x-h)^2 +k ; (h,k)#being vertex , we find here #h=-2,k=3# Hence vertex is at #(-2,3)#
y-intercept is obtained by putting #x=0# in the equation. #:. y=2(0+2)^2+3 =11# or at # (0,11) #
x-intercept is obtained by putting #y=0# in the equation. #:. 2(x+2)^2 +3=0 or 2(x+2)^2 = -3 or (x+2)^2 = -3/2# or #(x+2) = +- sqrt(-3/2) or x = -2 +- sqrt(-3/2)# or # x = -2 +- sqrt(3/2)i :. x# has complex roots. So there is no x-intercept.

graph{2(x+2)^2+3 [-40, 40, -20, 20]} [Ans]

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Answer 2

To find the vertex and intercepts for the quadratic function y = 2(x + 2)^2 + 3:

  1. Find the vertex:

    • The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex.
    • Comparing the given function to the vertex form, we can see that h = -2 and k = 3.
    • Therefore, the vertex is (-2, 3).
  2. Find the x-intercepts (zeros):

    • To find the x-intercepts, set y = 0 and solve for x.
    • 0 = 2(x + 2)^2 + 3
    • Solve for (x + 2)^2 = -3/2. Since a square cannot be negative, there are no real solutions, indicating no x-intercepts.
  3. Find the y-intercept:

    • To find the y-intercept, set x = 0 and solve for y.
    • y = 2(0 + 2)^2 + 3
    • y = 2(4) + 3
    • y = 8 + 3
    • y = 11
    • Therefore, the y-intercept is (0, 11).

The vertex is (-2, 3), and the y-intercept is (0, 11). There are no x-intercepts.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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