For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)(x-5)

Verified step by step guidance

1. Start by identifying the zeros of the polynomial function ƒ(x) = -(x-2)(x-5). The zeros are the values of x that make the function equal to zero. Set each factor equal to zero: x - 2 = 0 and x - 5 = 0, which gives zeros at x = 2 and x = 5.
2. Determine the end behavior of the polynomial. Since the function is a product of two linear factors and has a leading coefficient of -1 (negative), the degree is 2 (quadratic), and the parabola opens downward.
3. Find the vertex of the parabola. The vertex lies midway between the zeros, so calculate the midpoint: \( x = \frac{2 + 5}{2} = 3.5 \). Substitute this x-value back into the function to find the y-coordinate of the vertex.
4. Use the vertex and zeros to sketch the general shape of the graph: it crosses the x-axis at 2 and 5, and the vertex is the highest point since the parabola opens downward.
5. Compare this information with the given graph choices A–F, looking for a parabola that opens downward, crosses the x-axis at 2 and 5, and has its vertex at x = 3.5 with the corresponding y-value.

Key Concepts

Factored Form of a Polynomial

The factored form expresses a polynomial as a product of its factors, revealing its roots or zeros. For ƒ(x) = -(x-2)(x-5), the roots are x = 2 and x = 5, where the graph crosses the x-axis. Understanding this helps identify key points on the graph.

Leading Coefficient and End Behavior

The leading coefficient affects the graph's direction as x approaches infinity or negative infinity. Here, the leading coefficient is negative (due to the negative sign), so the parabola opens downward. This determines the overall shape and orientation of the graph.

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function. It can be found by averaging the roots (x = (2+5)/2 = 3.5) and evaluating ƒ at that x-value. Knowing the vertex helps in sketching the graph accurately.