Answer:

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To find the standard form of each function using completing the square, we can follow these steps:

1. f(x) = x^2-8x+5

First, we need to complete the square for the quadratic term. We can do this by adding and subtracting half of the coefficient of the x-term squared. In this case, the coefficient of the x-term is -8, so we have:

f(x) = (x^2-8x+16) - 16 + 5

Now, we can rewrite it as:

f(x) = (x-4)^2 - 11

The vertex of the parabola is at (4, -11), and the axis of symmetry is the vertical line x = 4.

2. f(x) = 1+3x+x^2

Similarly, we complete the square for the quadratic term:

f(x) = (x^2+3x+9/4) - 9/4 + 1

Rewriting it, we have:

f(x) = (x+3/2)^2 + 1/4

The vertex of the parabola is at (-3/2, 1/4), and the axis of symmetry is the vertical line x = -3/2.

3. f(x) = -x^2+x+1

Completing the square:

f(x) = -(x^2-x+1/4) + 1/4 + 1

Rewriting it:

f(x) = -(x-1/2)^2 + 5/4

The vertex of the parabola is at (1/2, 5/4), and the axis of symmetry is the vertical line x = 1/2.

4. f(x) = x^2+6x+8

Completing the square:

f(x) = (x^2+6x+9) - 9 + 8

Rewriting it:

f(x) = (x+3)^2 - 1

The vertex of the parabola is at (-3, -1), and the axis of symmetry is the vertical line x = -3.

Now, you can sketch the graph of each function using the information provided. Remember to label the vertex and axis of symmetry on each graph.