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If y=−12x2−9y=-\frac{1}{2}x^{2}-9 is graphed in the xyxy-plane, which of the following characteristics of the graph are displayed as a constant or coefficient in the equation?\newlineI. xx-intercept(s)\newlineII. yy-intercept\newlineIII. yy-coordinate of the vertex\newlineChoose 11 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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Q. If y=−12x2−9y=-\frac{1}{2}x^{2}-9 is graphed in the xyxy-plane, which of the following characteristics of the graph are displayed as a constant or coefficient in the equation?\newlineI. xx-intercept(s)\newlineII. yy-intercept\newlineIII. yy-coordinate of the vertex\newlineChoose 11 answer:\newline(A) II only\newline(B) III only\newline(C) I and II only\newline(D) II and III only
  1. Recognize Standard Form: The given quadratic equation is y=−(12)x2−9y = -\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=ax2+bx+cy = ax^2 + bx + c, where:\newline- aa is the coefficient that determines the direction and ＂width＂ of the parabola,\newline- bb is the coefficient related to the symmetry of the parabola and the xx-coordinate of the vertex,\newline- cc is the constant term that gives the yy-intercept of the graph.\newlineIn our equation, a=−(12)a = -\left(\frac{1}{2}\right), b=0b = 0, and c=−9c = -9.
  2. Coefficient aa Analysis: The coefficient a=−(12)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 xx-intercepts or the yy-coordinate of the vertex. It does influence the shape and direction of the parabola.
  3. Coefficient bb Analysis: Since b=0b = 0 in our equation, there is no term that directly affects the xx-coordinate of the vertex in the form of bxbx. This means that the vertex's xx-coordinate is not directly given by the equation in its current form. However, the absence of bb indicates symmetry about the yy-axis.
  4. Constant Term cc Analysis: The constant term c=−9c = -9 is the y-intercept of the graph. This is because when x=0x = 0, y=cy = c. Therefore, the point (0,−9)(0, -9) is where the graph intersects the y-axis.
  5. Vertex Calculation: The yy-coordinate of the vertex in a quadratic equation in the form y=ax2+bx+cy = ax^2 + bx + c can be found using the formula −b2a-\frac{b}{2a} for the xx-coordinate of the vertex and then substituting this back into the equation to find the yy-coordinate. However, since b=0b = 0 in our equation, the xx-coordinate of the vertex is 00, and substituting x=0x = 0 into the equation gives us the yy-coordinate of the vertex, which is y=ax2+bx+cy = ax^2 + bx + c00. Therefore, the yy-coordinate of the vertex is directly given by the constant term in the equation.
  6. Characteristics Summary: Based on the analysis:\newline- The xx-intercepts are not directly given by the coefficients or constants in the equation.\newline- The yy-intercept is directly given by the constant term c=−9c = -9.\newline- The yy-coordinate of the vertex is also directly given by the constant term c=−9c = -9, as shown in the previous step.\newlineTherefore, the characteristics of the graph displayed as a constant or coefficient in the equation are the yy-intercept and the yy-coordinate of the vertex.

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