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).
$a = - 4$
$a = 4$
$a = 1$

Explanation:

Step1: Analyze the sign of \(a\)

For a parabola \(y = ax^2+bx + c\), if \(a>0\), the parabola opens upward; if \(a < 0\), it opens downward. The first graph (left - most) opens downward, so its \(a\) value should be negative. So \(a=- 4\) matches the left - most graph.

Step2: Analyze the width of the parabola

The width of the parabola \(y = ax^2+bx + c\) is related to the absolute value of \(a\). The larger the absolute value of \(a\), the narrower the parabola. For \(a = 4\) and \(a=1\), since \(|4|>|1|\), the parabola with \(a = 4\) is narrower than the one with \(a = 1\). The middle graph is narrower than the right - most graph. So \(a = 4\) matches the middle graph and \(a=1\) matches the right - most graph.

Answer:

• \(a=-4\): The left - most graph (opening downward).
• \(a = 4\): The middle graph (narrower upward - opening parabola).
• \(a=1\): The right - most graph (wider upward - opening parabola).