Question

assuming all parabolas are of the form $y = ax^2 + bx + c$, drag and drop the graphs to match the appropriate a - value (if necessary).
$a = -0.25$
$a = -4$
$a = -1$
answer attempt 1 out of 3

Explanation:

Step1: Recall parabola properties

For a parabola \( y = ax^2+bx + c \), the sign of \( a \) determines direction (negative opens down), and the magnitude of \( |a| \) determines width: larger \( |a| \) means narrower, smaller \( |a| \) means wider.

Step2: Analyze \( |a| \) magnitudes

Given \( a=-0.25 \), \( a = - 4 \), \( a=-1 \). The magnitudes are \( | - 0.25|=0.25 \), \( |-4| = 4 \), \( |-1|=1 \). So \( 0.25<1<4 \), meaning \( a = - 0.25 \) gives the widest parabola, \( a=-4 \) the narrowest, \( a = - 1 \) in between.

Step3: Match graphs to \( a \)-values

• The widest (since \( | - 0.25| \) is smallest) opening downward: first graph (left - most) matches \( a=-0.25 \).
• The narrowest (since \( | - 4| \) is largest) opening downward: third graph (right - most) matches \( a=-4 \).
• The middle - width ( \( |-1| = 1 \)) opening downward: second graph (middle) matches \( a=-1 \)? Wait, no, re - check: Wait, the second graph (middle) seems to have a different vertex? Wait, no, all are downward opening (since \( a<0 \)). Wait, the first graph (left) is a wider parabola, the middle graph: wait, maybe I mixed up. Wait, \( |a| \)越大，抛物线越窄。So \( a=-0.25 \) (|a|=0.25) is widest, \( a=-1 \) (|a|=1) is middle, \( a=-4 \) (|a|=4) is narrowest. So:
• Widest (a=-0.25): first graph (left)
• Middle (a=-1): middle graph? Wait, no, the third graph (right) is narrowest. Wait, the three graphs: left graph: wider, middle graph: maybe? Wait, no, let's re - express:
• For \( a=-0.25 \): widest (since 0.25 is smallest |a|), so the left - most graph (first) with the widest curve.
• For \( a=-1 \): middle width (|a|=1), so the middle graph? Wait, no, the right - most graph is the narrowest (|a|=4), so:
• \( a=-0.25 \): left graph (widest)
• \( a=-1 \): middle graph? Wait, no, the middle graph's shape: wait, maybe the left graph (first) is \( a=-0.25 \) (widest), middle graph is \( a=-1 \) (middle width), right graph is \( a=-4 \) (narrowest).

So:

• \( a=-0.25 \) → first graph (left)
• \( a=-1 \) → middle graph
• \( a=-4 \) → third graph (right)

Answer:

• \( a = - 0.25 \) matches the left - most graph.
• \( a=-1 \) matches the middle graph.
• \( a=-4 \) matches the right - most graph.