If $y=-\frac{1}{2}x^{2}-9$ is graphed in the $xy$ -plane, which of the following characteristics of the graph are displayed as a constant or coefficient in the equation? $\newline$ I. $x$ -intercept(s) $\newline$ II. $y$ -intercept $\newline$ III. $y$ -coordinate of the vertex $\newline$ Choose $1$ answer: $\newline$ (A) II only $\newline$ (B) III only $\newline$ (C) I and II only $\newline$ (D) II and III only

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Algebra 2

Find the vertex of the transformed function

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Q. If

y=-\frac{1}{2}x^{2}-9

is graphed in the

xy

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

\newline

x

-intercept(s)

\newline

II.

y

-intercept

\newline

III.

y

-coordinate of the vertex

\newline

Choose

1

answer:

\newline

(A) II only

\newline

(B) III only

\newline

\newline

(D) II and III only

Recognize Standard Form: The given quadratic equation is $y = -\left(\frac{1}{2}\right)x^2 - 9$ . To understand which characteristics of the graph are displayed as a constant or coefficient in the equation, we first need to recognize the standard form of a quadratic equation, which is $y = ax^2 + bx + c$ , where: $\newline$ - $a$ is the coefficient that determines the direction and ＂width＂ of the parabola, $\newline$ - $b$ is the coefficient related to the symmetry of the parabola and the $x$ -coordinate of the vertex, $\newline$ - $c$ is the constant term that gives the $y$ -intercept of the graph. $\newline$ In our equation, $a = -\left(\frac{1}{2}\right)$ , $b = 0$ , and $c = -9$ .
Coefficient $a$ Analysis: The coefficient $a = -(\frac{1}{2})$ determines the direction (since it's negative, the parabola opens downwards) and the ＂width＂ of the parabola but does not directly give us the $x$ -intercepts or the $y$ -coordinate of the vertex. It does influence the shape and direction of the parabola.
Coefficient $b$ Analysis: Since $b = 0$ in our equation, there is no term that directly affects the $x$ -coordinate of the vertex in the form of $bx$ . This means that the vertex's $x$ -coordinate is not directly given by the equation in its current form. However, the absence of $b$ indicates symmetry about the $y$ -axis.
Constant Term $c$ Analysis: The constant term $c = -9$ is the y-intercept of the graph. This is because when $x = 0$ , $y = c$ . Therefore, the point $(0, -9)$ is where the graph intersects the y-axis.
Vertex Calculation: The $y$ -coordinate of the vertex in a quadratic equation in the form $y = ax^2 + bx + c$ can be found using the formula $-\frac{b}{2a}$ for the $x$ -coordinate of the vertex and then substituting this back into the equation to find the $y$ -coordinate. However, since $b = 0$ in our equation, the $x$ -coordinate of the vertex is $0$ , and substituting $x = 0$ into the equation gives us the $y$ -coordinate of the vertex, which is $y = ax^2 + bx + c$ $0$ . Therefore, the $y$ -coordinate of the vertex is directly given by the constant term in the equation.
Characteristics Summary: Based on the analysis: $\newline$ - The $x$ -intercepts are not directly given by the coefficients or constants in the equation. $\newline$ - The $y$ -intercept is directly given by the constant term $c = -9$ . $\newline$ - The $y$ -coordinate of the vertex is also directly given by the constant term $c = -9$ , as shown in the previous step. $\newline$ Therefore, the characteristics of the graph displayed as a constant or coefficient in the equation are the $y$ -intercept and the $y$ -coordinate of the vertex.