The shape of quadratic graphs is mostly decided by the numbers (called coefficients) in a special kind of math equation. This equation looks like this:

$$f(x) = ax^2 + bx + c$$

Let’s break down what each part means.

1. Coefficient 'a':

• Which Way It Opens:
  The number 'a' tells us if the graph (which is shaped like a U) opens up or down:
  • If $a > 0$: The graph opens up. This means the highest point is in the valley (the vertex).
  • If $a < 0$: The graph opens down. This means the highest point is at the top of the hill (the vertex).
• How Wide It Is:
  • If 'a' is a big number (e.g., $|a| > 1$), the U shape is narrow.
  • If 'a' is a small number (e.g., $|a| < 1$), the U shape is wide. For example, $f(x) = 2x^2$ (narrow) is different from $f(x) = 0.5x^2$ (wide).

2. Coefficient 'b':

• Where the Vertex Is:
  The number 'b' changes where the middle point (the vertex) of the graph is along the side (the x-axis). We can find out the exact spot using this formula:

$$x = -\frac{b}{2a}$$

When 'b' changes, the place where 'x' is located moves left or right.

• How Steep or Shallow It Is:
  The value of 'b' also affects how steep the graph is, but not as much as 'a' does.

3. Coefficient 'c':

• How High or Low the Graph Is:
  The number 'c' shows where the graph crosses the vertical line (the y-axis). When 'c' changes, it moves the whole graph up or down, but it does not change the shape of the U.

Conclusion:

Knowing how the numbers 'a', 'b', and 'c' in the equation $f(x) = ax^2 + bx + c$ affect the shape and position of the graph is very important. This helps us understand and draw curved lines. By changing these numbers, we can guess how the graph will look and where it will go. This is key for 8th-grade math, especially when working with curved functions.