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Understanding the Graphs and Coefficients:

Each graph represents a quadratic function of the form $y = Ax^2$, $y = Bx^2$, $y = Cx^2$, and $y = Dx^2$. The coefficient (A, B, C, or D) determines:

1. The direction of the parabola (upward or downward).
2. How wide or narrow the parabola is.

(a) Determining if Each Coefficient is Positive or Negative:

• A positive coefficient leads to an upward-opening parabola.
• A negative coefficient leads to a downward-opening parabola.

By analyzing the graphs:

• The first two graphs (for $A$ and $B$) are upward-opening, meaning $A > 0$ and $B > 0$.
• The last two graphs (for $C$ and $D$) are downward-opening, meaning $C < 0$ and $D < 0$.

(b) Coefficient Closest to Zero:

The parabola closest to a straight line (or the widest parabola) corresponds to the coefficient that is closest to zero because smaller coefficients make the parabola wider.

• The second graph (for $B$) has the widest parabola, so $B$ is closest to zero.

(c) Coefficient with the Greatest Value:

The coefficient with the narrowest parabola (the most vertical) will have the greatest value because larger coefficients make the parabola narrower.

• The first graph (for $A$) has the narrowest parabola, so $A$ has the greatest value.

Final Answers:

• (a) Coefficient signs:
  • $A$: Positive
  • $B$: Positive
  • $C$: Negative
  • $D$: Negative
• (b) Closest to 0: $B$
• (c) Greatest value: $A$

Would you like further details on any part of this explanation?

Related Questions:

1. What effect does changing the coefficient have on the width of the parabola?
2. How does the sign of the coefficient affect the direction of the parabola?
3. What happens to the graph when the coefficient equals zero?
4. How would the graphs change if the coefficients were fractions?
5. Can the vertex of these parabolas change if we alter the coefficient?

Tip:

Remember, larger absolute values of coefficients make the parabola narrower, and smaller absolute values make it wider.