Questions: Choose the parabola with the narrowest graph. y=-4 x^2 y=1/2 x^2 y=-2 x^2 y=-1/3 x^2

Transcript text: Choose the parabola with the narrowest graph. $y=-4 x^{2}$ $y=\frac{1}{2} x^{2}$ $y=-2 x^{2}$ $y=-\frac{1}{3} x^{2}$

Solution

Solution Steps

To determine which parabola has the narrowest graph, we need to compare the coefficients of $x^2$ in each equation. The narrower the parabola, the larger the absolute value of the coefficient. We will convert all equations to the form $y = ax^2$ and then compare the absolute values of $a$.

Step 1: Identify the Parabolas

The given parabolas are:

$ 9y = -4x^2 $ which can be rewritten as $ y = -\frac{4}{9}x^2 $
$ y = \frac{1}{2}x^2 $
$ y = -2x^2 $
$ y = -\frac{1}{3}x^2 $

Step 2: Determine the Coefficients

The coefficients of $ x^2 $ for each parabola are:

For $ y = -\frac{4}{9}x^2 $, the coefficient is $ -\frac{4}{9} $
For $ y = \frac{1}{2}x^2 $, the coefficient is $ \frac{1}{2} $
For $ y = -2x^2 $, the coefficient is $ -2 $
For $ y = -\frac{1}{3}x^2 $, the coefficient is $ -\frac{1}{3} $

Step 3: Calculate Absolute Values

The absolute values of the coefficients are:

$ \left| -\frac{4}{9} \right| = \frac{4}{9} \approx 0.4444 $
$ \left| \frac{1}{2} \right| = \frac{1}{2} = 0.5 $
$ \left| -2 \right| = 2 $
$ \left| -\frac{1}{3} \right| = \frac{1}{3} \approx 0.3333 $

Step 4: Identify the Narrowest Parabola

The largest absolute value among the coefficients is $ 2 $, which corresponds to the parabola $ y = -2x^2 $. This indicates that this parabola has the narrowest graph.