Question

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Answer 1

Answer:

(i) The axis of symmetry is $x = -4$.

(ii) The axis of symmetry is $x = 4$.

(iii) The axis of symmetry is $x = 1$.

Step-by-step explanation:

We proceed to explain how we determine the expression for the axis of symmetry for each case:

(i) $f(x) = 3\cdot (x+4)^{2}+1$

The expression depicts a vertical parabola, meaning that axis of symmetry is vertical, that is, a function of the form $x = a,\,\forall\,a\in \mathbb{R}$. This line passes through the vertex of the parabola. The component $x+4$ contains the information of the horizontal component of the vertex, where the axis passes through. Therefore, the axis of symmetry is $x = -4$.

(ii) $f(x) = 2\cdot x^{2} -16\cdot x +15$

At first we complete the square and factor the perfect square trinomial:

$f(x) = 2\cdot \left(x^{2}-8\cdot x +\frac{15}{2} \right)$

$f(x) = 2\cdot \left[(x^{2}-8\cdot x +16) +\frac{15}{2}-16 \right]$

$f(x) = 2\cdot (x-4)^{2}-17$

By applying the approach used in (i), we find that the axis of symmetry is $x = 4$.

(iii) First, we locate the vertex of the parabola, which is (1,-3). The first component of the ordered pair contains all the needed information to determine the equation of the axis of symmetry, which is $x = 1$.