Let’s analyze the given problem and answer each part step by step:

Given Polynomial:

$f(x) = x^4 + x^3 - 28x^2 + 20x + 48$

Part (a): Why do the ends of this polynomial point in the same direction?

The polynomial $f(x)$ is a quartic (degree 4) polynomial. For a polynomial, the behavior of the ends is determined by the leading term. Here, the leading term is $x^4$, which has a positive coefficient. Quartic polynomials with positive leading terms have their ends pointing in the same direction (upward). This is unlike cubic polynomials, whose odd-degree nature causes their ends to point in opposite directions.

Part (b): Transformation to make the ends point downward without changing the zeros

To make the ends point downward, we need to negate the leading term of $f(x)$. This can be achieved by multiplying the entire polynomial by $-1$, giving: $f(x) \to -f(x) = -(x^4 + x^3 - 28x^2 + 20x + 48).$ This transformation flips the graph vertically, reversing the end behavior, but it does not affect the zeros of the polynomial.

Part (c): Circle the four real zeros and list them

From the graph, the four real zeros appear to be approximately at: $x = -6, \, x = -2, \, x = 2, \, x = 4.$

Part (d): How vertically shifting the function could remove real zeros

By vertically shifting the graph of $f(x)$, we add or subtract a constant to/from the polynomial: $f(x) + k,$ where $k$ is a constant.

• Effect of vertical shifting on zeros: Adding a constant $k$ can move the entire graph up or down. If $k$ is large enough (positive or negative), the graph may no longer intersect the x-axis. When this happens, the polynomial would have no real zeros.

For instance, shifting $f(x)$ downward (subtracting a large $k$) could cause the local maxima and minima to drop below the x-axis, eliminating all real roots.

Would you like to explore the answers in more detail or see specific calculations? Here are related questions for deeper understanding:

1. How does the degree of a polynomial influence the end behavior of its graph?
2. What transformations can preserve the shape of the graph but affect the direction of its ends?
3. How can we algebraically confirm the approximate zeros using synthetic division or the Rational Root Theorem?
4. What happens to the number of real zeros if we vertically stretch or compress the graph?
5. Can you describe the role of the discriminant in identifying the nature of the roots?

Tip: When analyzing transformations of functions, always focus on how they affect key features like intercepts, zeros, and end behavior!