Evaluating F(t) = 2t^2 - 18t + 26 And Finding The Maximum Height

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Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically focusing on the function f(t) = 2t^2 - 18t + 26. We'll explore how to evaluate this function for different values of t, and more importantly, we'll figure out how to determine its maximum value. So, buckle up and let's get started!

Understanding Quadratic Functions

First things first, let's quickly recap what quadratic functions are all about. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable is 2. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards.

In our case, we have f(t) = 2t^2 - 18t + 26. Notice that the coefficient of the t^2 term (which is a) is 2, which is positive. This tells us that the parabola opens upwards, meaning the function has a minimum value, not a maximum. However, the question mentions "maximum height," which might be a bit misleading in this context. We'll address this later when we discuss applications.

The key features of a parabola are its vertex, axis of symmetry, and intercepts. The vertex is the point where the parabola changes direction – it's either the lowest point (minimum) or the highest point (maximum) on the graph. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The intercepts are the points where the parabola intersects the x-axis (x-intercepts or roots) and the y-axis (y-intercept).

Understanding these fundamental concepts is crucial for effectively working with quadratic functions. Now, let's move on to evaluating our specific function, f(t) = 2t^2 - 18t + 26, for different values of t.

Evaluating f(t) = 2t^2 - 18t + 26 for Different Values

To evaluate a function, we simply substitute the given value of the variable (in this case, t) into the function's expression and simplify. Let's try a few examples:

  • f(0): Substitute t = 0 into the function:

    • f(0) = 2(0)^2 - 18(0) + 26 = 0 - 0 + 26 = 26

  • f(1): Substitute t = 1 into the function:

    • f(1) = 2(1)^2 - 18(1) + 26 = 2 - 18 + 26 = 10

  • f(2): Substitute t = 2 into the function:

    • f(2) = 2(2)^2 - 18(2) + 26 = 8 - 36 + 26 = -2

  • f(5): Substitute t = 5 into the function:

    • f(5) = 2(5)^2 - 18(5) + 26 = 50 - 90 + 26 = -14

  • f(9): Substitute t = 9 into the function:

    • f(9) = 2(9)^2 - 18(9) + 26 = 162 - 162 + 26 = 26

By substituting different values of t, we can see how the function's output, f(t), changes. This gives us a sense of the function's behavior and how it curves. We can plot these points on a graph to visualize the parabola. The process of evaluating the function for various values of t helps us to understand the shape and position of the parabola on the coordinate plane. It is an important step in understanding the overall characteristics and behavior of any given function.

Now that we've practiced evaluating the function, let's move on to the more interesting question of finding its maximum (or minimum) value. Remember, since the parabola opens upwards, we're actually looking for the minimum value in this case. We will now delve into methods that are used to determine the maximum or minimum value of such functions, focusing on the mathematical approach that will help us reach a precise conclusion.

Determining the Minimum Value of f(t) = 2t^2 - 18t + 26

There are a couple of ways to find the minimum value of a quadratic function. One method involves completing the square, and the other uses the vertex formula. Let's explore both.

Method 1: Completing the Square

Completing the square is a technique that rewrites a quadratic expression in the form a(x - h)^2 + k, where (h, k) are the coordinates of the vertex. In our case, we want to rewrite f(t) = 2t^2 - 18t + 26 in this form. Here's how:

  1. Factor out the coefficient of the t^2 term from the first two terms:

    • f(t) = 2(t^2 - 9t) + 26

  2. Complete the square inside the parentheses: To complete the square, we need to add and subtract the square of half the coefficient of the t term. The coefficient of the t term is -9, so half of it is -9/2, and the square of that is (-9/2)^2 = 81/4.

    • f(t) = 2(t^2 - 9t + 81/4 - 81/4) + 26

  3. Rewrite the expression inside the parentheses as a perfect square:

    • f(t) = 2((t - 9/2)^2 - 81/4) + 26

  4. Distribute the 2 and simplify:

    • f(t) = 2(t - 9/2)^2 - 2(81/4) + 26
    • f(t) = 2(t - 9/2)^2 - 81/2 + 26
    • f(t) = 2(t - 9/2)^2 - 81/2 + 52/2
    • f(t) = 2(t - 9/2)^2 - 29/2

Now, the function is in the form f(t) = a(t - h)^2 + k, where a = 2, h = 9/2, and k = -29/2. The vertex of the parabola is at the point (h, k) = (9/2, -29/2). Since the parabola opens upwards, the vertex represents the minimum point, and the minimum value of the function is k = -29/2 = -14.5.

Method 2: Using the Vertex Formula

The vertex formula provides a direct way to find the coordinates of the vertex of a parabola. For a quadratic function in the form f(x) = ax^2 + bx + c, the x-coordinate of the vertex (which we'll call h) is given by:

  • h = -b / 2a

In our case, f(t) = 2t^2 - 18t + 26, so a = 2 and b = -18. Plugging these values into the formula, we get:

  • h = -(-18) / (2 * 2) = 18 / 4 = 9/2

To find the y-coordinate of the vertex (which we'll call k), we substitute h back into the function:

  • k = f(9/2) = 2(9/2)^2 - 18(9/2) + 26
    • k = 2(81/4) - 81 + 26
    • k = 81/2 - 81 + 26
    • k = 81/2 - 162/2 + 52/2
    • k = -29/2 = -14.5

Again, we find that the vertex is at the point (9/2, -29/2), and the minimum value of the function is -29/2 = -14.5. This aligns with the result we obtained using the completing the square method. Either of these methods provides a mathematically sound way of finding the minimum value, and the consistency of the results increases our confidence in the conclusion.

Both methods confirm that the minimum value of the function f(t) = 2t^2 - 18t + 26 is -14.5, which occurs at t = 9/2 = 4.5.

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