Point 1 — Quadratics and parabolas

What do we see?

Vertex, roots, or coefficient combinations of a quadratic given indirectly.

What do we understand?

We see?	We understand?
f(a) = f(b)	x-coordinate of the vertex = $\frac{a + b}{2}$
Number of unknown coefficients > number of points on the graph	Discriminant, or the y-coordinate of the vertex
a+b+c / a−b+c / 4a−2b+c [any combination of a, b, c]	There is some x-value that generates this combination as its corresponding y-value
A quadratic equation has factors (px − m) where p and m are integers	Real roots, and the discriminant is greater than or equal to 0
Roots of a quadratic equation given	Average of the roots = the vertex’s x-coordinate
Increasing then decreasing around point (h, k) or vice versa	Vertex is (h, k)
Any information about the vertex	Use − $\frac{b}{2 a}$ to solve for a or b if possible, or to establish a relationship
Any relation between a+b or a+c	Use the sum or product formulas, or plug in a certain point

Question 1 of 8

f(x) = ax² + 18x + c

In the given quadratic function, a and c are constants. The graph of y = f(x) in the xy-plane is a parabola that opens upward and has a vertex at the point (h, k), where h and k are constants. If k > 0 and f(−5) = f(2), which of the following must be true?

I. c > 16

II. a > 1