Eleventh Grade · Math · week 8

Quadratic functions in vertex form

Graph a quadratic function given in vertex form and identify its vertex, direction of opening, and width compared to the parent parabola.

Every parabola is a transformed version of the same parent, f(x)=x^2 -- which is why last month's transformation rules apply directly to quadratics. Vertex form, f(x) = a(x-h)^2 + k, packs the whole graph into three numbers: (h,k) is the vertex (the parabola's turning point), and a controls both the direction it opens and how narrow or wide it is.

If a is positive, the parabola opens upward and the vertex is a MINIMUM; if a is negative, it opens downward and the vertex is a MAXIMUM. The larger |a| is, the narrower (steeper) the parabola; the smaller |a| is (between 0 and 1), the wider (flatter) it is -- exactly the vertical stretch/compression rule from two weeks ago, applied to the parabola family specifically.

📋 For the teacher
Teach it (10–15 min):
  • Point out the sign flip explicitly every time: in (x-3)^2 the vertex's x-coordinate is POSITIVE 3, not negative -- this trips up almost everyone at first.
  • Have them sort a handful of vertex-form equations into "opens up" and "opens down" piles by sight alone, before graphing anything.
  • Compare |a| values side by side (like a=1, a=3, a=1/3 with the same vertex) to build real intuition for narrow versus wide.
  • Always confirm with one plugged-in point (like f(5) in the example) -- a fast, reliable check that the vertex and the rest of the curve agree.
Talk about it:
  • Why does (x - 3)^2 shift the parabola RIGHT when the number inside looks positive?
  • What is the minimum or maximum value of a parabola in vertex form, and where do you read it?
  • How would the graph change if a went from 2 to 1/2, keeping the same vertex?

Mastery looks like: They read the vertex and direction of opening directly from vertex form, describe the parabola's width, and evaluate the function at a given point.

Common stumbles: Reading the vertex's x-coordinate with the wrong sign; confusing which number (a, h, or k) controls width versus position.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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