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.