Ninth Grade · Math · week 25

Quadratic functions: the parabola's anatomy

Graph quadratic functions from standard form, identifying the y-intercept, axis of symmetry, and vertex.

Every function you graphed in term one made a straight line. A quadratic, y = ax² + bx + c, makes a curve called a parabola - a smooth U-shape (or upside-down U) caused entirely by that squared term. The sign of a decides which way it opens: positive a opens upward like a bowl (with a lowest point), negative a opens downward like a dome (with a highest point).

Standard form hands you two facts for free: c is the y-intercept (plug in x = 0 and everything with an x disappears), and the sign of a tells you the direction. The vertex - the parabola's tip, its minimum or maximum - sits on the axis of symmetry, the vertical line x = -b/(2a) that splits the parabola into two mirror-image halves. Find that x-value, substitute it back in for the matching y, and you have located the single most important point on the graph.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Sketch a bowl and a dome side by side and physically label which sign of a makes each - the picture should come before any formula.
  • Say the axis-of-symmetry formula out loud as "the opposite of b, over twice a" until the words are automatic, then explain WHY it locates the middle: it is the x-value exactly between any two mirror-image points.
  • Have them verify the vertex two ways on one example: substitute the axis x-value, AND check that the two x-intercepts really are equally spaced around it.
  • Keep c and the y-intercept explicitly linked - "set x to zero" should be an instant reflex by the end of the week.
Talk about it:
  • Why does the sign of a decide whether the parabola opens up or down?
  • Why is the vertex always exactly on the axis of symmetry, never off to one side?
  • What does the y-intercept tell you that the vertex does not?

Mastery looks like: They can find the y-intercept, axis of symmetry, and vertex of any y = ax² + bx + c, and state the opening direction from a alone.

Common stumbles: Forgetting the negative sign in x = -b/(2a); mixing up the vertex's x-coordinate with the y-intercept.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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