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).
$a = - 4$
$a = - 1$
$a = 0.25$

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 opens upward, so \(a>0\). Among the given \(a\)-values, only \(a = 0.25\) is positive. So the first graph matches \(a=0.25\).
• The second and third graphs open downward, so \(a<0\).

Step2: Analyze the magnitude of \(|a|\)

The magnitude of \(a\) determines the width of the parabola. The larger \(|a|\) is, the narrower the parabola.

• For \(a=- 4\) and \(a = - 1\), \(|-4|=4\) and \(|-1| = 1\). Since \(4>1\), the parabola with \(a=-4\) is narrower than the one with \(a=-1\).
• The second graph is narrower than the third graph (when opening downward, a larger \(|a|\) makes it narrower). So the second graph (narrower downward opening) matches \(a=-4\), and the third graph (wider downward opening) matches \(a=-1\).

Answer:

• \(a = 0.25\): The first graph (opening upward)
• \(a=-4\): The second graph (narrower downward opening)
• \(a = - 1\): The third graph (wider downward opening)