Math

Graphing Parabolas

2 min readMedium5-question drill

Parabolas show up all over the Math section — projectile paths, profit curves, area problems — and the test loves to ask for the vertex, the x-intercepts, or where the graph hits its max or min. Knowing the three forms of a quadratic lets you read those answers straight off the equation.

Where this lesson fits

Graphing Parabolas

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A parabola is the U-shaped curve you get when you graph a quadratic function — any function with an x² term as its highest power. The standard way to write one is:

f(x) = ax² + bx + c

The number a controls direction and width. If a > 0, the parabola opens upward (like a smile, with a lowest point). If a < 0, it opens downward (like a frown, with a highest point).

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f(x) = x² − 6x + 4 opens upward (a > 0) with its vertex (minimum) at (3, −5).

The single most important point is the vertex — the turning point where the curve changes direction. On an upward parabola it's the minimum; on a downward parabola it's the maximum. The test asks for this constantly.

Three forms of a quadratic, and what each one hands you for free:

Standard form: f(x) = ax² + bx + c. The c value is the y-intercept (where the graph crosses the y-axis).
Vertex form: f(x) = a(x − h)² + k. The vertex is (h, k) — read it directly. Watch the sign: (x − 3) means h = 3, but (x + 3) means h = −3.
Factored form: f(x) = a(x − p)(x − q). The x-intercepts (also called roots or zeros) are x = p and x = q.

Finding the vertex from standard form is a core skill. The x-coordinate of the vertex is always:

x = −b / (2a)

Plug that x-value back into the function to get the y-coordinate. For example, with f(x) = x² − 4x − 1: x = −(−4)/(2·1) = 2, then f(2) = 4 − 8 − 1 = −5, so the vertex is (2, −5).

The axis of symmetry is the vertical line x = −b/(2a) that runs through the vertex. The parabola is a mirror image on either side of it.

Knowing which form you're given tells you which features are already visible — and which formula you need to dig out the rest.

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Quick check

What is the vertex of the parabola f(x) = -x² + 2x + -5?

(-1, -4)

(1, 4)

(1, -4)

(2, -5)

Worked examples

Example 1

What is the vertex of the parabola f(x) = x² − 6x + 4?

Example 2

The function f(x) = −(x + 2)² + 9 is graphed in the xy-plane. What is the maximum value of f(x)?

Example 3

If f(x) = 2x² − 3x − 5, what is the value of f(3)?

Common pitfalls

Flipping the sign of h in vertex form

Vertex form is a(x − h)² + k, so (x + 4)² means h = −4, NOT 4. The vertex's x-coordinate is the value that makes the parenthesis equal zero — always read the opposite sign.

Forgetting to substitute back for the vertex's y-value

x = −b/(2a) only gives the x-coordinate. Students stop there and pick an answer, but you must plug that x back into f(x) to get the y-coordinate of the vertex.

Confusing the y-intercept with the vertex

In standard form ax² + bx + c, the c is the y-intercept, not the vertex's y-value. They're only the same when b = 0.

Dropping a negative when squaring during f(x) evaluation

When you plug a negative number in, square it before applying any coefficient: 2(−3)² = 2(9) = 18, not −18. Use parentheses to keep the sign straight.

Key takeaways

A parabola opens up when a > 0 (has a minimum) and down when a < 0 (has a maximum).
The vertex x-coordinate from standard form is x = −b/(2a); substitute back to find the y-coordinate.
Vertex form a(x − h)² + k hands you the vertex (h, k) directly — watch the sign of h.
Factored form a(x − p)(x − q) hands you the x-intercepts x = p and x = q.
Evaluating f(number) just means substituting and following order of operations carefully.

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Try it yourself

5 practice questions on Graphing Parabolas, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Graphing Parabolas

You are here

→Nonlinear Functions →Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables →Equivalent Expressions