Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(h(t)=-\frac{2}{3}\left(t^{2}-5 t+3\right)\)

Step-by-Step Solution

Answer

The right-hand and left-hand behavior of the graph of the function \(h(t)\) both point downwards.

Step 1: Identify the leading coefficient and degree of the polynomial

The given function \(h(t) = -\frac{2}{3}(t^{2}-5t+3)\) is a quadratic function. The leading coefficient is -2/3, and the degree of the polynomial is 2, which is an even number.

Step 2: Apply the Leading Coefficient Test

In the Leading Coefficient Test, if the polynomial has an even degree and a positive leading coefficient, both ends of the graph point upwards. If the degree is even and the leading coefficient is negative, both ends point downwards. As in this case, the leading coefficient is -2/3 (which is negative) and the degree is 2 (which is even), according to the Leading coefficient test, both ends of the graph will point downwards.

Step 3: Verify with a graphing utility

To verify our result, we can use a graphing utility. When we graph the function \(h(t) = -\frac{2}{3}(t^{2}-5t+3)\), we can see that as 't' increases or decreases, the value of \(h(t)\) decreases. The graph opens downwards, which confirms that both ends of the graph point downwards.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a straightforward yet powerful tool that helps us predict the end behavior of a polynomial function's graph. When we talk about "end behavior," we're interested in the directions the graph tends towards as the value of the variable becomes very large, positively or negatively.

For this, we look at two things: the degree of the polynomial and the leading coefficient, which is the coefficient of the term with the highest power.

• If the degree is even and the leading coefficient is positive, both ends of the graph will point upward.
• If the degree is even and the leading coefficient is negative, like in our function, both ends will point downward.
• When the degree is odd and the leading coefficient is positive, the end on the right will point upward while the end on the left goes downward.
• Conversely, if the degree is odd and the leading coefficient is negative, the right end will point downward, and the left will point upward.

In our example, because the leading coefficient is -2/3 (negative) and the degree is 2 (even), we know from the Leading Coefficient Test that both ends of our quadratic graph point downward.

Quadratic Function

A quadratic function is a second-degree polynomial, meaning the highest exponent of the variable is 2. The standard form of a quadratic function looks like this: \[ y = ax^2 + bx + c \] where "a," "b," and "c" are constants, and "a" is non-zero.

Quadratic functions are known for their characteristic bell-shaped curves called parabolas. These parabolas can open upwards or downwards depending on the sign of the leading coefficient "a."

• If "a" is positive, the parabola opens upwards, creating a U-shape.
• If "a" is negative, as in our example, the parabola opens downward, resembling an inverted U.

Our function \( h(t) = -\frac{2}{3}(t^2 - 5t + 3) \) is indeed a quadratic one. Since we identified our leading coefficient "a" as -2/3, our parabola will open downward. The vertex of the parabola, a point that represents its peak when it opens downward, plays a crucial role in determining the function's minimum or maximum value.

Graphing Utility Verification

Graphing utilities are fantastic tools that offer a visual representation of algebraic functions. They provide an intuitive way to check the results predicted analytically. When dealing with polynomial behavior, plotting the graph with a utility allows us to verify our conclusions with a visual aid.

With the function \( h(t) = -\frac{2}{3}(t^2 - 5t + 3) \), using a graphing utility, we expect to see a parabola that opens downwards. As we have discussed using the Leading Coefficient Test, ends of the graph should point downwards.

Once plotted, the graph shows how \( h(t) \) behaves as "t" increases or decreases. As expected, the graph dips down on both sides, confirming the analysis both through the test and real graphing. Such verification provides confidence in algebraic solutions and boosts understanding by showcasing the mathematics visually.