Question: Look at the graphs and their equations below. Then fill in the information about the leading coefficients $A$, $B$, $C$, and $D$. y = $Ax^2$ y = $Bx^2$ y = $Cx^2$ y = $Dx^2$ (a) For each coefficient, choose whether it is positive or negative. $A$: (Choose one) $B$: (Choose one) $C$: (Choose one) $D$: (Choose one) (b) Choose the coefficient closest to 0. $A$ $B$ $C$ $D$ (c) Choose the coefficient with the greatest value. $A$ $B$ $C$ $D$ Graph 1: a narrow parabola opening downward with vertex at the origin, position_hint = top-left. Graph 2: a wider parabola opening downward with vertex at the origin, position_hint = top-right. Graph 3: a narrow parabola opening upward with vertex at the origin, position_hint = center. Graph 4: a wider parabola opening upward with vertex at the origin, position_hint = top-right.

1. **State the problem:** We have four parabolas given by $y = Ax^2$, $y = Bx^2$, $y = Cx^2$, and $y = Dx^2$. We need to determine for each coefficient $A$, $B$, $C$, and $D$ whether it is positive or negative, which coefficient is closest to zero, and which has the greatest value. 2. **Recall the properties of parabolas:** - If the parabola opens upward, the leading coefficient is positive. - If the parabola opens downward, the leading coefficient is negative. - The "narrowness" of the parabola relates to the absolute value of the coefficient: a larger absolute value means a narrower parabola. 3. **Analyze each graph:** - Graph 1: narrow parabola opening downward $ ightarrow A < 0$ and $|A|$ is large. - Graph 2: wider parabola opening downward $ ightarrow B < 0$ and $|B|$ is smaller than $|A|$. - Graph 3: narrow parabola opening upward $ ightarrow C > 0$ and $|C|$ is large. - Graph 4: wider parabola opening upward $ ightarrow D > 0$ and $|D|$ is smaller than $|C|$. 4. **Answer (a) sign of coefficients:** - $A$: negative - $B$: negative - $C$: positive - $D$: positive 5. **Answer (b) coefficient closest to zero:** - Since $B$ and $D$ are wider parabolas, their coefficients have smaller absolute values. - Among $B$ and $D$, both are closer to zero than $A$ and $C$. - We cannot distinguish which is closer without exact numbers, but both $B$ and $D$ are candidates. - Since $B$ and $D$ are both wider, the coefficient closest to zero is either $B$ or $D$. 6. **Answer (c) coefficient with greatest value:** - Among $A$, $B$, $C$, and $D$, the greatest value is the largest positive number. - $C$ is positive and narrow (large positive), $D$ is positive and wider (smaller positive). - So $C$ has the greatest value. **Final answers:** (a) $A$: negative $B$: negative $C$: positive $D$: positive (b) Coefficient closest to zero: $B$ or $D$ (wider parabolas) (c) Coefficient with greatest value: $C$