Q. what does the y-intercept represent

What the y-intercept represents — detailed step-by-step explanation

1. Definition (what to look for).

   The y-intercept is the point where a graph crosses the y-axis. On the coordinate plane the y-axis is the set of points whose x-coordinate is zero, so the y-intercept has the form
   \( (0, y_0) \)
   where \( y_0 \) is the y-value at that crossing.

2. How to find it from slope-intercept form (what to do).

   If a line is given in slope-intercept form
   \( y = m x + b \),
   the constant \( b \) is the y-intercept. To see this, substitute \( x = 0 \):
   \[
   y = m\cdot 0 + b = b.
   \]
   Therefore the y-intercept point is \( (0, b) \). So the practical step: identify the equation in the form \( y = m x + b \) and read off \( b \).

3. How to find it from standard form (what to do).

   If a line is given in standard form \( A x + B y = C \), set \( x = 0 \) and solve for \( y \):
   \[
   A\cdot 0 + B y = C \quad\Rightarrow\quad B y = C \quad\Rightarrow\quad y = \frac{C}{B},
   \]
   provided \( B \neq 0 \). The y-intercept point is \( \left(0, \dfrac{C}{B}\right) \). So the practical step: plug \( x=0 \) into the equation and solve for \( y \).

4. How to find it from a graph (what to do).

   Look for the point where the curve or line crosses the vertical axis (the y-axis). Read the y-coordinate of that crossing. That coordinate is the y-intercept value \( y_0 \), and the point is \( (0, y_0) \).

5. Interpretation in context (what it means).

   In application problems where \( y \) depends on \( x \), the y-intercept represents the value of \( y \) when the independent variable \( x \) equals zero. Commonly this is called the initial value or starting amount. Examples:

   • If \( y \) is money and \( x \) is time in months, the y-intercept is the amount of money at time zero (the initial balance).
   • If \( y \) is height and \( x \) is time, the y-intercept is the height at time zero.

   For instance, given \( y = 5x + 3 \), the y-intercept is \( 3 \) and the point is \( (0,3) \); this means when \( x = 0 \), \( y = 3 \).

6. Special cases to note (what to watch for).
   • If the y-intercept is \( 0 \) the graph passes through the origin, point \( (0,0) \).
   • A vertical line \( x = c \) with \( c \neq 0 \) does not have a y-intercept because it never crosses the y-axis. The vertical line \( x = 0 \) is the y-axis itself and intersects at every \( y \).
   • Horizontal lines \( y = k \) have y-intercept \( k \) and the point \( (0,k) \).

Summary statement: The y-intercept is the point on the graph where x equals zero, usually written \( (0,y_0) \); algebraically it is the value of the function when \( x=0 \), and in context it represents the initial or starting value of the quantity represented by \( y \).