The graph of $y=(x-2)(x-4)$ is shown in the $x y$ -plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation as written? $\newline$ Choose $1$ answer:

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Q. The graph of

y=(x-2)(x-4)

is shown in the

x y

-plane. Which of the following characteristics of the graph is displayed as a constant or coefficient in the equation as written?

\newline

Choose

1

answer:

Identifying Roots: The equation $y=(x-2)(x-4)$ is in factored form, which directly shows the roots of the polynomial. The roots are the $x$ -values where the graph intersects the $x$ -axis. To find the roots, we set $y$ to zero and solve for $x$ . $\newline$ $0 = (x-2)(x-4)$ $\newline$ This gives us the roots $x = 2$ and $x = 4$ .
Graph Intercepts: The roots of the polynomial are the x-intercepts of the graph. These are the points where the graph crosses the x-axis. In the given equation, the roots are explicitly shown as $x = 2$ and $x = 4$ , which correspond to the factors $(x-2)$ and $(x-4)$ , respectively.
Constant Term: The constant term in the expanded form of the polynomial is the product of the roots when all the roots are negative, or it is the negative of the product when the roots have different signs. However, since we are not asked to expand the polynomial, we do not need to calculate the constant term in this step. Instead, we focus on the characteristics displayed in the given factored form.
Coefficients: The coefficients in the equation as written are the numbers in front of the $x$ terms inside the parentheses, which are $-2$ and $-4$ . These coefficients are related to the roots of the polynomial, but they are not the characteristics we are looking for since the question asks for the characteristic displayed as a constant or coefficient in the equation as written.
Constant Characteristic: The constant characteristic in the equation $y=(x-2)(x-4)$ as written is not explicitly shown because the equation is in factored form. To see the constant term, we would need to expand the equation, which is not required for this question. Therefore, the characteristic displayed as a constant in the equation is not visible in its current form.