Find the $y$-intercept.
$\begin{array}{c}
y=\frac{4 x+12}{4 x-12} \
(0,[?])
\end{array}$
Answer (2)
To find the y -intercept, substitute x = 0 into the equation.
Simplify the expression: y = 4 ( 0 ) − 12 4 ( 0 ) + 12 = − 12 12 .
Calculate the value of y : y = − 1 .
The y -intercept is − 1 .
Explanation
Understanding the Problem We are given the equation y = 4 x − 12 4 x + 12 and asked to find the y -intercept. The y -intercept is the point where the graph of the equation intersects the y -axis. This occurs when x = 0 .
Substituting x=0 To find the y -intercept, we substitute x = 0 into the equation: y = 4 ( 0 ) − 12 4 ( 0 ) + 12 y = 0 − 12 0 + 12 y = − 12 12 y = − 1
Finding the y-intercept Therefore, the y -intercept is the point ( 0 , − 1 ) . The y -coordinate of the y -intercept is − 1 .
Examples
Understanding y-intercepts is crucial in many real-world applications. For instance, in business, if you graph the cost of producing items versus the number of items, the y-intercept represents the fixed costs—the costs you have even if you produce nothing. Similarly, in physics, if you graph the distance an object travels over time, the y-intercept could represent the object's initial position. Knowing how to find and interpret y-intercepts helps in making informed decisions and understanding initial conditions in various scenarios.
To find the y -intercept of the equation y = 4 x − 12 4 x + 12 , substitute x = 0 into the equation, which simplifies to y = − 1 . Therefore, the y -intercept is the point ( 0 , − 1 ) .
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