Which statements are true about the graph of the function f(x) = 6x - 4 + x²?
(i) The vertex form of the function is f(x) = (x - 2)² + 2.
(ii) The vertex of the function is (-3, -13).
(iii) The axis of symmetry for the function is x = 3.
(iv) The graph increases over the interval (-3, infinity).
(v) The function does not cross the x-axis.

Solution:

Given: Function f(x) = 6x - 4 + x²

Rearranging the function in the general form, we get

⇒ y = x² +6x - 4 

The standard form of the quadratic function is of the form y = (x- h)²+k

The vertex (h,k) of a parabola is found by completing the square.

y = x² +6x - 4

y = (x + 3)² -4 -9 = (x + 3)² -13

thus the function becomes y = (x + 3)² - 13

Statement (i) is false because, f(x) = (x + 3)² -13

Statement (ii) is true because, f(x) = (x + 3)² -13 can also be represented as (x + 3)² = y + 13 comparable to (x - h)² = 4a (y - k) a rotated parabola whose vertex is (h, k) Thus, vertex of given function is (-3, -13)

Statement (iii) is false because the axis of symmetry is x = -3. The axis of symmetry intersects the parabola at x axis.

Statement (iv) is true (x + 3)² = y + 13 is a parabola which opens upwards hence, the graph increases over interval (-3, ∞).

Statement (v) is false.

When y = 0;

(x + 3)² = 0 + 13

(x + 3)² = 13

x + 3 = ± √13

x = -3 ± √13

⇒ Curve cuts x-axis at 2 points -3 + √13 or -3 - √13.

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Which statements are true about the graph of the function f(x) = 6x - 4 + x²?
(i) The vertex form of the function is f(x) = (x - 2)² + 2.
(ii) The vertex of the function is (-3, -13).
(iii) The axis of symmetry for the function is x = 3.
(iv) The graph increases over the interval (-3, infinity).
(v) The function does not cross the x-axis.

Summary:

For the function f(x) = 6x - 4 + x², the vertex of the function is (-3, -13), the graph increases over the interval (-3, infinity).