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Math  /  Algebra

QuestionFill in the information about the parabolas below. (a) For each parabola, choose whether it opens upward or downward. y=−x2: (Choose one) vy=−13x2: (Choose one) vy=12x2: (Choose one) v (Choose one) vy=-x^{2}: \text { (Choose one) } v y=-\frac{1}{3} x^{2}: \text { (Choose one) } v \quad y=\frac{1}{2} x^{2}: \text { (Choose one) } v \text { (Choose one) } v (b) Choose the parabola with the narrowest graph. y=−x2y=-x^{2} y=−13x2y=-\frac{1}{3} x^{2} y=12x2y=\frac{1}{2} x^{2} y=−3x2y=-3 x^{2} (c) Choose the parabola with the widest graph. y=−x2y=-x^{2} y=−13x2y=-\frac{1}{3} x^{2} y=12x2y=\frac{1}{2} x^{2} y=−3x2y=-3 x^{2}

Studdy Solution

STEP 1

1. A parabola in the form y=ax2 y = ax^2 opens upward if a>0 a > 0 and downward if a<0 a < 0 .
2. The "narrowness" or "wideness" of a parabola is determined by the absolute value of the coefficient a a . A larger absolute value indicates a narrower graph, while a smaller absolute value indicates a wider graph.

STEP 2

1. Determine the direction each parabola opens.
2. Identify the parabola with the narrowest graph.
3. Identify the parabola with the widest graph.

STEP 3

Determine the direction each parabola opens by examining the sign of the coefficient a a .
- For y=−x2 y = -x^2 , a=−1 a = -1 . Since a<0 a < 0 , it opens downward. - For y=−13x2 y = -\frac{1}{3}x^2 , a=−13 a = -\frac{1}{3} . Since a<0 a < 0 , it opens downward. - For y=12x2 y = \frac{1}{2}x^2 , a=12 a = \frac{1}{2} . Since a>0 a > 0 , it opens upward.

STEP 4

Identify the parabola with the narrowest graph by comparing the absolute values of the coefficients a a .
- For y=−x2 y = -x^2 , ∣a∣=1|a| = 1. - For y=−13x2 y = -\frac{1}{3}x^2 , ∣a∣=13|a| = \frac{1}{3}. - For y=12x2 y = \frac{1}{2}x^2 , ∣a∣=12|a| = \frac{1}{2}. - For y=−3x2 y = -3x^2 , ∣a∣=3|a| = 3.
The parabola with the largest ∣a∣|a| is y=−3x2 y = -3x^2 , so it has the narrowest graph.

STEP 5

Identify the parabola with the widest graph by comparing the absolute values of the coefficients a a .
The parabola with the smallest ∣a∣|a| is y=−13x2 y = -\frac{1}{3}x^2 , so it has the widest graph.
(a) Directions: - y=−x2 y = -x^2 : downward - y=−13x2 y = -\frac{1}{3}x^2 : downward - y=12x2 y = \frac{1}{2}x^2 : upward
(b) Narrowest graph: y=−3x2 y = -3x^2
(c) Widest graph: y=−13x2 y = -\frac{1}{3}x^2

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