Question

The function f(t) = t2 + 4t - 14 represents a parabola.

Part A: Rewrite the function in vertex form by completing the square. Show your
work.

Part B: Determine the vertex and indicate whether it is a maximum or a minimum on
the graph. How do you know?

Part C: Determine the axis of symmetry for f(t).

Answer 1

Answer:

a: $f(t)=(t+2)^2-18$

b: vertex: (-2, -18)

   minimum

c: x = -2

Step-by-step explanation:

Completing the square is difficult to explain in words. but I'll try my best. Let me know if you need any more information.

Part A ===============

Start with the given function in this form:

$f(x)=ax^2+bx+c\\t^2+4t-14$

We need an additional number, d, and we'll calculate it like this:

$d=(\frac{b}{2a})^2 \\\\d=(\frac{4}{2(1)} )^2\\d=2^2\\d=4$

d needs to be added and subtracted at the same time. That way we do not change the value of the expression:

$ax^2+bx+d-d+c\\\\t^2+4t+4-4-14$

And you can separate them like this to make it easier to read:

$(t^2+4t+4) + (-4-14)\\(t^2+4t+4)-18$

Finally, we need to factor the trinomial in the parenthesis. Because we've completed the square, it's now a perfect square trinomial.  The factored form is just:

$(x+e)^2$

where e is half of b, or the square root of c. That looks like this in our case, and you can also expand it to make sure.

$(t+2)^2\\\\(t+2)(t+2)\\t^2+2t+2t+4\\t^2+4t+4$

Put that all together and you get the answer:

$f(t)=(t+2)^2-18$

Part B ===============

Vertex form is this:

$f(x)=a(x-h)^2+k$

where (h, k) is the vertex and a is the scale factor.

In part A, we converted the given function into vertex form. Looking at that, you can see the vertex is at (-2, -18), and the scale factor is 1. When the scale factor is positive, the parabola faces upwards. When it's negative, the parabola faces downwards.

The scale factor is a positive 1 in this case, and if you picture that in your mind, the vertex is at the very lowest point, or the minimum, of the parabola.

Part C ==============

The axis of symmetry is always on the vertex, and that is because the vertex is the exact point where the parabola changes direction from positive to negative, or negative to positive.

The vertex of this parabola is (-2, -18), so the axis of symmetry is at x = -2.

Sorry if that got too long and hard to follow, I can clarify any part of it if needed.