Math  /  Algebra

Question1). Find the coordinates of the vertex of the following quadratic functions: [3K] b).. y=4x216x+7y=4 x^{2}-16 x+7

Studdy Solution

STEP 1

1. The given function y=4x216x+7 y = 4x^2 - 16x + 7 is a quadratic function.
2. The vertex form of a quadratic function is y=a(xh)2+k y = a(x-h)^2 + k , where (h,k)(h, k) is the vertex.
3. The vertex can also be found using the formula h=b2a h = -\frac{b}{2a} for the standard form ax2+bx+c ax^2 + bx + c .

STEP 2

1. Identify coefficients a a , b b , and c c .
2. Use the vertex formula to find the x-coordinate of the vertex.
3. Substitute the x-coordinate back into the function to find the y-coordinate.
4. Write the vertex as an ordered pair.

STEP 3

Identify the coefficients from the quadratic function y=4x216x+7 y = 4x^2 - 16x + 7 :
- a=4 a = 4 - b=16 b = -16 - c=7 c = 7

STEP 4

Use the vertex formula h=b2a h = -\frac{b}{2a} to find the x-coordinate of the vertex:
h=162×4 h = -\frac{-16}{2 \times 4} h=168 h = \frac{16}{8} h=2 h = 2

STEP 5

Substitute x=2 x = 2 back into the original function to find the y-coordinate:
y=4(2)216(2)+7 y = 4(2)^2 - 16(2) + 7 y=4(4)32+7 y = 4(4) - 32 + 7 y=1632+7 y = 16 - 32 + 7 y=16+7 y = -16 + 7 y=9 y = -9

STEP 6

Write the vertex as an ordered pair:
The vertex of the quadratic function is (2,9)(2, -9).
The coordinates of the vertex are:
(2,9) \boxed{(2, -9)}

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