Question

The function
`f(t) = t^2 + 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

Part A: To rewrite the function in vertex form by completing the square, let's start with a general quadratic function:

f(t) = at^2 + bt + c

To complete the square, we need to focus on the t-term (bt) and the constant term (c).

Step 1: Divide the coefficient of the t-term by 2 and square it:
(b/2)^2 = (b^2) / 4

Step 2: Add and subtract the value obtained in Step 1 inside the parentheses:
f(t) = at^2 + bt + (b^2) / 4 - (b^2) / 4 + c

Step 3: Rearrange the terms:
f(t) = a(t^2 + (b/2a)t + (b^2) / (4a)) + (c - (b^2) / (4a))

Now, we have a perfect square trinomial inside the parentheses. We can simplify it further:
f(t) = a(t + (b/2a))^2 + (c - (b^2) / (4a))

Part B: The vertex form of a parabola is given by f(t) = a(t - h)^2 + k, where (h, k) represents the vertex coordinates. By comparing this with our rewritten function, we can identify the vertex.

In our case, the vertex is (-b/2a, c - (b^2) / (4a)). The x-coordinate of the vertex is -b/2a, and the y-coordinate is c - (b^2) / (4a).

Depending on the coefficient 'a', the parabola will have either a maximum or minimum.

If 'a' is positive, the parabola opens upward, and the vertex represents the minimum point on the graph.

If 'a' is negative, the parabola opens downward, and the vertex represents the maximum point on the graph.

Part C: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is given by the equation t = -b/2a.

So, the axis of symmetry for f(t) is t = -b/2a.

By applying these steps, you can find the vertex form, determine the vertex, and identify the axis of symmetry for any given quadratic function. Keep up the math-tastic work!

Answer 2

Look below

Explanation:

Part A:

To rewrite the function in vertex form by completing the square, we need to follow these steps:

1. Start with the quadratic function in standard form: f(t) = at^2 + bt + c

2. Identify the coefficients a, b, and c from the given function.

3. Divide the coefficient of t by 2 and square the result to find the constant to complete the square. Let's call this constant k.

4. Add and subtract k inside the parentheses after the term involving t.

5. Group the terms involving t and factor out the common factor in that group.

6. Simplify the equation and rewrite it in vertex form.

Part B:

To determine the vertex and whether it is a maximum or minimum on the graph, we can use the vertex form of the quadratic function, which is f(t) = a(t - h)^2 + k.

The vertex of the parabola is represented by the point (h, k). By comparing the equation in vertex form with the given function, we can identify the values of h and k.

If the coefficient a is positive, the parabola opens upward and the vertex represents the minimum point on the graph. If the coefficient a is negative, the parabola opens downward and the vertex represents the maximum point on the graph.

Part C:

The axis of symmetry for f(t) is a vertical line that passes through the vertex of the parabola. It can be found using the formula x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry is represented by the equation t = h.

By finding the vertex in Part B and identifying the x-coordinate of the vertex, you can determine the equation for the axis of symmetry for f(t).