Question

the graph of which function has an axis of symmetry at x = 3? f(x) = x2 3x 1 f(x) = x2 – 3x – 3 f(x) = x2 6x 3 f(x) = x2 – 6x – 1

Answer 1

A parabola has an axis of symmetry at x = 3 if the x-value of the vertex is 3.

f(x) = x^2 + 3x + 1 = x^2 + 3x + 9/4 - 5/4 = (x + 3/2)^2 - 5/4 => vertex = (-3/2, -5/4)
f(x) = x^2 - 3x - 3 = x^2 - 3x + 9/4 - 21/4 = (x - 3/2)^2 - 21/4 => vertex = (3/2, -21/4)
f(x) = x^2 + 6x + 3 = x^2 + 6x + 9 - 6 = (x + 3)^2 - 6 => vertex = (-3, -6)
f(x) = x^2 - 6x - 1 = x^2 - 6x + 9 - 10 = (x - 3)^2 - 10 => vertex = (3, -10)

Therefore, f(x) = x^2 - 6x - 1 has an axis of symmetry at x = 3.

Answer 2

For this case, the function that has an axis of symmetry is given by:

$f (x) = x ^ 2 - 6x - 1$

To verify this, what we must do is to derive the function, because the axis of symmetry passes through the minimum or maximum point of the function.

We have then:

$f '(x) = 2x - 6$

From here, we equate the function to zero:

$2x - 6 = 0$

Then, we clear the value of x.

We have then:

$2x = 6$

$x =\frac{6}{2} x = 3$

Answer:

A function that has an axis of symmetry at x = 3 is:

$f (x) = x ^ 2 - 6x - 1$