A downward-facing parabola passes through the points (1,7) ,(3,13), and (9,7) .
Find b if the parabola passes through (7,b) .
The equation of a downward-facing parabola is of the form y=a(x−h)2+k, where a is the coefficient of the squared term, h is the horizontal shift of the parabola, and k is the vertical shift of the parabola.
We can find h and k by substituting the known points into the equation. For example, if we substitute the point (1,7), we get:
We can solve this quadratic equation for h to get:
Substituting the point (3,13), we get:
Substituting the point (9,7), we get:
Substituting a=−1, h=3, and k=4−3a into the equation y=a(x−h)2+k, we get:
Therefore, if the parabola passes through the point (7,b), then b=−72+6(7)−2=−13.