Problem 51 - Algebra A Combined Function

Question

The graph of each equation is an ellipse. Determine which distance is longer, the distance between the \(x\) -intercepts or the distance between the y-intercepts. How much longer? $$ 4 x^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

The y-intercept distance is 4 units longer than the x-intercept distance.

Step 1: Identify the Standard Form of the Ellipse Equation

The equation given is \(4x^2 + y^2 = 16\). We will rewrite it in the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Divide every term by 16 to convert the equation: \(\frac{x^2}{4} + \frac{y^2}{16} = 1\). This shows that \(a^2 = 4\) and \(b^2 = 16\).

Step 2: Calculate the x-intercepts

For x-intercepts, set \(y = 0\) in the standard form equation: \(\frac{x^2}{4} + \frac{0^2}{16} = 1\) simplifies to \(\frac{x^2}{4} = 1\). Solving for \(x\) gives \(x^2 = 4\), so \(x = \pm 2\). The x-intercepts are (-2, 0) and (2, 0).

Step 3: Calculate the Distance Between x-intercepts

The distance between the x-intercepts (-2, 0) and (2, 0) is calculated as the difference between the x-values: \(2 - (-2) = 4\).

Step 4: Calculate the y-intercepts

For y-intercepts, set \(x = 0\) in the standard form equation: \(\frac{0^2}{4} + \frac{y^2}{16} = 1\) simplifies to \(\frac{y^2}{16} = 1\). Solving for \(y\) gives \(y^2 = 16\), so \(y = \pm 4\). The y-intercepts are (0, -4) and (0, 4).

Step 5: Calculate the Distance Between y-intercepts

The distance between the y-intercepts (0, -4) and (0, 4) is calculated as the difference between the y-values: \(4 - (-4) = 8\).

Step 6: Compare the Distances and Determine the Difference

The distance between the x-intercepts is 4, and the distance between the y-intercepts is 8. The y-intercept distance is longer. To find how much longer, subtract the x-intercept distance from the y-intercept distance: \(8 - 4 = 4\).

Key Concepts

Understanding x-intercepts

An ellipse is a shape that can be graphed on an x-y coordinate plane, much like a circle, but elongated.
To find where this ellipse touches or crosses the x-axis, we identify its x-intercepts.
This is the point(s) where the value of the y-coordinate is zero.

• Start by setting \( y = 0 \) in the ellipse equation \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \).
• This simplifies to \( \frac{x^2}{4} = 1 \), leading us to solve for \( x \).

By solving \( x^2 = 4 \), we find \( x = \pm 2 \). So, the x-intercepts are at (-2, 0) and (2, 0).
The x-intercepts are the points where the ellipse meets the horizontal x-axis.

Exploring y-intercepts

Y-intercepts mark where the shape crosses the y-axis, meaning the x-coordinate at these points is zero.
To determine the y-intercepts, we substitute \( x = 0 \) into the ellipse equation \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \).

• This reduces to \( \frac{y^2}{16} = 1 \), from which we solve \( y^2 = 16 \).
• Taking the square root gives \( y = \pm 4 \).

Thus, the y-intercepts occur at the points (0, -4) and (0, 4).
These are the locations along the y-axis where our ellipse touches or intersects.

Calculating distances

When determining the length between intercepts on an ellipse, we look at the differences in their coordinates.
For the x-intercepts (-2, 0) and (2, 0), use the formula for distance along a straight line:
The distance is simply the difference along the x-axis, which is \(2 - (-2) = 4\).

• This tells us the x-intercepts are 4 units apart.

Similarly, for the y-intercepts (0, -4) and (0, 4), we calculate their distance:

• The difference in the y-coordinates is \(4 - (-4) = 8\).
• So, the y-intercepts are 8 units apart.

Conclusively, since 8 is greater than 4, the distance between the y-intercepts is longer.
We see that the y-intercepts are 4 units further apart than the x-intercepts.