Finding Maximum Value of y in an Ellipse Equation

The given equation is $4x^2 + y^2 = 9$. We are asked to find the maximum value that the variable $y$ can take.

This equation represents an ellipse centered at the origin. To understand the maximum possible values for $x$ and $y$, we can analyze the equation. The term $4x^2$ is always greater than or equal to zero ($4x^2 \ge 0$), and similarly, the term $y^2$ is always greater than or equal to zero ($y^2 \ge 0$).

The equation $4x^2 + y^2 = 9$ tells us that the sum of $4x^2$ and $y^2$ is always equal to 9.

Calculating the Maximum Value of y

To find the maximum value of $y$, we need to consider the condition under which $y^2$ is maximized. Since $4x^2$ is always non-negative, $y^2$ will be maximized when $4x^2$ is minimized. The minimum possible value for $4x^2$ is 0, which occurs when $x = 0$.

Let's substitute $x = 0$ into the equation:

$\qquad 4(0)^2 + y^2 = 9$

$\qquad 0 + y^2 = 9$

$\qquad y^2 = 9$

To find the values of $y$, we take the square root of both sides:

$\qquad y = \pm\sqrt{9}$

$\qquad y = \pm 3$

This gives us two possible values for $y$ when $x=0$: $y = 3$ and $y = -3$.

The values of $y$ can range from -3 to 3. The maximum value among these is 3.

Alternatively, we can rearrange the equation to express $y^2$ in terms of $x^2$:

$\qquad y^2 = 9 - 4x^2$

For $y$ to be a real number, $y^2$ must be non-negative. Also, since $4x^2 \ge 0$, the term $9 - 4x^2$ will be largest when $4x^2$ is smallest. The smallest value of $4x^2$ is 0 (when $x=0$).

Maximum value of $y^2 = 9 - 0 = 9$.

So, $y^2 \le 9$. Taking the square root, we get $|y| \le 3$. This means $-3 \le y \le 3$.

The maximum value $y$ can attain is 3.

Analyzing the Options for Maximum y

Let's look at the given options:

• Option 1: $\frac{3}{2}$
• Option 2: 3
• Option 3: 4
• Option 4: 6

Based on our calculation, the maximum value of $y$ is 3. This matches Option 2.

The analysis confirms that the range of $y$ is $[-3, 3]$, making the maximum value 3.

Revision Table: Understanding Maximum Values

Additional Information: Ellipse Geometry

The equation $4x^2 + y^2 = 9$ describes an ellipse centered at the origin (0,0).

• The points where the ellipse crosses the x-axis occur when $y=0$. $4x^2 + 0^2 = 9 \implies 4x^2 = 9 \implies x^2 = \frac{9}{4} \implies x = \pm \frac{3}{2}$. These points are $(\frac{3}{2}, 0)$ and $(-\frac{3}{2}, 0)$. The semi-major or semi-minor axis along the x-axis has length $\frac{3}{2}$.
• The points where the ellipse crosses the y-axis occur when $x=0$. $4(0)^2 + y^2 = 9 \implies y^2 = 9 \implies y = \pm 3$. These points are $(0, 3)$ and $(0, -3)$. The semi-major or semi-minor axis along the y-axis has length 3.

Since the length along the y-axis (3) is greater than the length along the x-axis ($\frac{3}{2}$), the major axis is along the y-axis. The vertices of the ellipse are at $(0, \pm 3)$ and the co-vertices are at $(\pm \frac{3}{2}, 0)$.

The maximum value of $y$ is the positive y-intercept, which is 3. The minimum value of $y$ is the negative y-intercept, which is -3.

Similarly, the maximum value of $x$ is the positive x-intercept, which is $\frac{3}{2}$. The minimum value of $x$ is the negative x-intercept, which is $-\frac{3}{2}$.