Question

1. which relation is not a function?

a. $3x^2 + 4y^2 = 12$
b. $y = 2x + 4$
c. $\frac{1}{2}x + y = 2$
d. $y = 3x^2 - 4$

Explanation:

To determine which relation is not a function, we use the vertical line test: a relation is a function if no vertical line intersects its graph more than once.

Step 1: Analyze Option B ($y = 2x + 4$)

This is a linear equation in slope - intercept form ($y=mx + b$). For every value of $x$, there is exactly one value of $y$. So, it passes the vertical line test and is a function.

Step 2: Analyze Option C ($\frac{1}{2}x + y=2$)

We can rewrite this as $y=-\frac{1}{2}x + 2$, which is also a linear equation. For each $x$, there is a unique $y$. It passes the vertical line test and is a function.

Step 3: Analyze Option D ($y = 3x^{2}-4$)

This is a quadratic function. The graph of a quadratic function is a parabola. A vertical line will intersect the parabola at most once. So, it passes the vertical line test and is a function.

Step 4: Analyze Option A ($3x^{2}+4y^{2}=12$)

We can rewrite this equation by dividing both sides by 12: $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, which is the equation of an ellipse. If we draw a vertical line through the ellipse, it will intersect the ellipse at two points (except at the vertices). So, it fails the vertical line test and is not a function.

Answer:

A. $3x^{2}+4y^{2}=12$