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An x-intercept of a polynomial function is a point on the graph where the curve crosses the x-axis. At these points, the y-value is zero because the graph is exactly at the level of the x-axis. To determine the x-intercepts of a factored polynomial, such as \( P(x) = (2-x)(x+5) \), we can set each factor equal to zero and solve for \( x \). This is because a product is zero only if at least one of its factors is zero.
For the given polynomial, the factors are \( (2-x) \) and \( (x+5) \).

• Setting \( 2-x = 0 \) gives \( x = 2 \).
• Setting \( x+5 = 0 \) gives \( x = -5 \).

Hence, the x-intercepts are at \( x = 2 \) and \( x = -5 \). These points mean that the graph passes through \( (2, 0) \) and \( (-5, 0) \), forming crucial parts of the graph's structure.

The y-intercept is where the graph crosses the y-axis, and it's determined by evaluating the function at \( x = 0 \). This value tells us where the polynomial will intersect the y-axis, giving insight into how the graph behaves in the vicinity of the origin.
For the polynomial \( P(x) = (2-x)(x+5) \), substitute \( x = 0 \) into the expression.

• Calculate \( P(0) = (2-0)(0+5) = 2 \times 5 = 10 \).

Thus, the y-intercept is the point \( (0, 10) \) on the graph. Knowing this helps in plotting the graph accurately, especially when combined with x-intercepts, as it anchors the graph on the y-axis.

End behavior in polynomials refers to what happens to the function's values as \( x \) approaches positive or negative infinity. For a more accurate sketch of the graph, understanding end behavior ensures that we know how the graph behaves "at the edges."
The given polynomial \( P(x) = (2-x)(x+5) \) can be expanded to give a leading term. When expanded, \( (2-x)(x+5) = -x^2 + 10 \) (ignoring lower degree terms for end behavior). The dominant term is \( -x^2 \), meaning as \( x \) moves towards infinity or minus infinity, \( P(x) \) heads towards negative infinity. This tells us that:

• As \( x \to +\infty \), \( P(x) \to -\infty \).
• As \( x \to -\infty \), \( P(x) \to -\infty \).

Essentially, the graph falls (heads downwards) at both ends, indicating a "downward" direction on both sides of the graph.

Graph sketching is about visually representing a polynomial function on a coordinate plane. It involves plotting the intercepts and considering the end behavior to complete the picture.
To sketch \( P(x) = (2-x)(x+5) \), start by plotting the intercepts identified earlier:

• X-intercepts at \( (2, 0) \) and \( (-5, 0) \).
• Y-intercept at \( (0, 10) \).

Once these points are plotted, the curve should pass smoothly through them, respecting the end behavior which indicates that both ends of the graph should fall downwards.
Therefore, connect the intercepts with a smooth curve, ensuring the graph falls as \( x \to \pm \infty \). The careful specification of these attributes results in a more accurate and clear visual representation of the polynomial function.