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If $y=(x-1)(x+5)$ is graphed in the $xy$ -plane, which of the following characteristics of the graph is displayed as a constant in the equation?

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Full solution

Q. If

y=(x-1)(x+5)

is graphed in the

xy

-plane, which of the following characteristics of the graph is displayed as a constant in the equation?

Expand the equation: The equation $y=(x-1)(x+5)$ is a quadratic equation in standard form $y=ax^2+bx+c$ . To identify the constant characteristic, we need to expand the equation. $\newline$ $y = (x-1)(x+5)$ $\newline$ $= x(x+5) - 1(x+5)$ $\newline$ $= x^2 + 5x - x - 5$ $\newline$ $= x^2 + 4x - 5$
Identify constant term: Now that we have the expanded form of the equation, $y = x^2 + 4x - 5$ , we can identify the constant term in the quadratic equation. $\newline$ The constant term is the term without the variable $x$ , which is $-5$ in this case.
Constant term significance: The constant term in a quadratic equation does not affect the shape of the parabola but determines the $y$ -intercept of the graph. The $y$ -intercept is the point where the graph crosses the $y$ -axis, which occurs when $x=0$ .
Find y-intercept: Substituting $x=0$ into the equation $y = x^2 + 4x - 5$ , we find the y-intercept: $\newline$ $y = (0)^2 + 4(0) - 5$ $\newline$ $y = -5$
Graph interpretation: The $y$ -intercept, which is $-5$ , is the constant characteristic displayed in the equation when graphed in the $xy$ -plane. It is the point $(0, -5)$ on the graph.

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