Circles and Parabolas

• Write and interpret the standard circle equation (x − h)² + (y − k)² = r² given center and radius
• Identify the vertex of a parabola given vertex form y = a(x − h)² + k and given standard form y = ax² + bx + c
• Apply the transformation rules: f(x) + k shifts up, f(x − h) shifts right, −f(x) reflects over the x-axis, f(−x) reflects over the y-axis
• Complete the square to convert x² + y² + Dx + Ey + F = 0 into standard circle form
• Predict how changing a, h, or k in y = a(x − h)² + k reshapes the parabola

The circle equation (x − h)² + (y − k)² = r² has center (h, k) and radius r. The signs are the classic trap: if the equation is (x + 3)² + (y − 5)² = 16, the center is (−3, 5) and the radius is 4, not 16. The ACT loves to test this sign flip. If a circle is given in expanded form like x² + y² − 6x + 4y − 12 = 0, complete the square on x and y separately to recover center and radius. Parabolas come in two forms the ACT tests: standard form y = ax² + bx + c, where the vertex x-coordinate is −b/(2a), and vertex form y = a(x − h)² + k, where the vertex is (h, k). Vertex form is faster whenever the problem asks about the maximum or minimum. The vertex is the max (if a < 0) or min (if a > 0). The value of a controls width and direction: |a| > 1 is narrower, |a| < 1 is wider, a < 0 opens downward. Transformations follow consistent rules. Starting from y = f(x): adding outside the function (f(x) + k) shifts vertically. Up if k > 0, down if k < 0. Adding inside (f(x − h)) shifts horizontally, and the direction is counterintuitive: f(x − 3) shifts right by 3, not left. A negative outside (−f(x)) flips over the x-axis. A negative inside (f(−x)) flips over the y-axis. Multiplying outside by a constant c > 1 stretches vertically by a factor of c; replacing x with cx (c > 1) compresses horizontally by a factor of 1/c. A graphing calculator can verify any transformation: graph both forms and compare. Ellipses also appear on hard questions: (x − h)²/a² + (y − k)²/b² = 1 has center (h, k) and reaches a units left and right of the center and b units up and down, so its full width is 2a and its full height is 2b. Hyperbolas look like ellipses with a minus sign: (x − h)²/a² − (y − k)²/b² = 1 (or with the y term first) also has center (h, k), read the same way from (x − h) and (y − k), and its two asymptotes cross at that center, so 4(x + 3)² − 9(y − 1)² = 36 has its asymptotes meeting at (−3, 1). If a circle's center and one point on it are given, the radius is the distance between them: center (1, 2) through (4, 6) has r = √(3² + 4²) = 5, so the equation is (x − 1)² + (y − 2)² = 25. To name the transformation that maps one parabola onto another with the same a, track the vertex: y = (x − 2)² + 1 to y = (x + 3)² − 4 moves the vertex from (2, 1) to (−3, −4), a shift 5 left and 5 down. When a circle is tangent to an axis, its radius equals the distance from the center to that axis (a circle centered at (4, −6) and tangent to the x-axis has radius 6).

• In the circle equation, the signs inside the parentheses are opposite the center coordinates. (x − 3)² means the center has x = +3, not −3.
• To find a parabola's max or min quickly, use x = −b/(2a) and plug back in; for a parabola it is usually faster than completing the square (save completing the square for circles).
• f(x − h) shifts right when h is positive. The minus sign is baked into the form. The sign of h is what matters.
• If the ACT asks for the radius of a circle written in expanded form, complete the square. Do not guess from the coefficients.
• A graphing calculator can verify this: graph the original and the transformed equation and compare the shift.