Coordinate Geometry • Topic 5 of 5

Parabolas

A parabola is the U-shaped graph of a quadratic, y = ax² + bx + c. The sign of a decides which way it opens: a > 0 opens upward (a valley with a lowest point), a < 0 opens downward (a hill with a highest point). That turning point is the vertex, and it is where the maximum or minimum output occurs — a frequent GRE target. In vertex form y = a(x − h)² + k the vertex is simply (h, k); otherwise the vertex's x-coordinate is x = −b/(2a). The x-intercepts (where y = 0) are exactly the solutions of the quadratic, so they connect straight back to factoring and the discriminant: two intercepts when b² − 4ac > 0, one when it equals 0, none when it is negative. The GRE stays graphical and conceptual here — expect "how many times does this parabola cross the x-axis?" or "is the vertex a maximum or a minimum?" rather than heavy computation.

An upward parabola in vertex form with its lowest point, the vertex, marked at (1, -3)Parabola: y = a(x − h)² + kxy-3-2-112345-3-2-112345Overtex (1, −3)a > 0: opens upvertex (h, k) = (1, −3); a = ½ > 0

✅ Solved examples

1. Does the parabola y = 2x² − 3x + 1 open upward or downward, and does its vertex give a max or a min?
a = 2 > 0, so it opens upward. An upward parabola has a lowest point, so the vertex is a minimum.
2. Find the x-coordinate of the vertex of y = x² − 6x + 5.
x = −b/(2a) = −(−6)/(2·1) = 6/2 = 3.
3. How many times does y = x² − 4x + 4 cross the x-axis?
Set y = 0: x² − 4x + 4 = (x − 2)² = 0. Discriminant is 0, so exactly one crossing (at x = 2, the vertex touches the axis).
4. For y = −(x − 3)² + 4, state the vertex and whether it is a maximum or minimum.
Vertex form gives vertex (3, 4). Since a = −1 < 0 the parabola opens downward, so (3, 4) is a maximum.

✏️ Practice — try these, take hints as needed

1. Does y = −3x² + x − 2 open upward or downward?
Look at the sign of a.
a = −3.
Negative a.
Downward
2. Find the x-coordinate of the vertex of y = x² + 8x + 1.
x = −b/(2a).
b = 8, a = 1.
−8/2.
x = −4
3. How many x-intercepts does y = x² + 2x + 5 have?
Use the discriminant b² − 4ac.
4 − 20.
Negative discriminant.
None (no real roots)
4. The vertex of y = (x + 2)² − 9 is at what point?
Vertex form is a(x − h)² + k.
x + 2 = x − (−2), so h = −2.
k = −9.
(−2, −9)
5. How many times does y = x² − 9 cross the x-axis?
Set y = 0.
x² − 9 = (x + 3)(x − 3).
Two distinct roots.
Twice (at x = 3 and x = −3)

📝 Topic test — 8 questions

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