The connection between the numbers in a quadratic equation and where the graph crosses the y-axis is really important to understand how the graph looks.

A common way to write a quadratic equation is:

$$y = ax^2 + bx + c$$

What is the Y-Intercept?

The y-intercept is where the graph meets the y-axis. This happens when $x = 0$.

If we put $x = 0$ into our equation, we get:

$$y = c$$

So, the y-intercept is simply the number $c$.

How Coefficients Affect the Graph

1. Effect of $c$ (The Constant)

   • The number $c$ decides how high or low the graph is on the y-axis.
   • If $c$ is positive, the graph goes up. If $c$ is negative, the graph goes down.
   • For example, if $c = 3$, the y-intercept is at the point $(0, 3)$. But if $c = -2$, the y-intercept is at $(0, -2)$.

2. Impact of $a$ (The Leading Number)

   • While $a$ doesn’t change where the graph crosses the y-axis, it does change the shape and direction of the curve.
   • If $a$ is positive, the curve opens upwards. If $a$ is negative, it opens downwards. This affects how we view the y-intercept compared to the highest or lowest point of the graph.

3. Role of $b$ (The Linear Number)

   • The number $b$ helps decide the balance and position of the peak or lowest point of the graph.
   • Even though $b$ doesn’t directly change where the graph meets the y-axis, it changes how the curve looks overall.

In Short

To sum it all up, the number $c$ in a quadratic equation shows us where the graph touches the y-axis. Meanwhile, numbers $a$ and $b$ change how the graph looks and moves, which helps us understand the y-intercept better. Knowing this is really helpful for looking at quadratic functions in Algebra I.