ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
Translating a Conic by Changing Its Equation
Replacing x with (x - h) and y with (y - k) in a conic's equation slides the whole curve to the point (h, k) without changing its size or shape.
Every conic equation you meet first — x2 + y2 = r2, y = ax2 — is centered or vertexed at the origin. To move it anywhere else, you don't redraw it, you just rename every point. Swap x for (x - h) and y for (y - k) inside the equation, and every ordered pair that used to satisfy the equation now has to shift h units right and k units up to satisfy the new one. The curve looks identical. Only its address changed.
The reason it feels backwards is the minus sign. Plugging in x = h makes (x - h) equal zero, the same job x = 0 did before, so the point that used to sit at the origin now sits at x = h. That's why y = a(x-4)2 has its vertex at x = 4, not x = -4, and why x2 + y2 = 16 becomes (x-3)2 + (y+2)2 = 16 after shifting 3 right and 2 down: the center moves from (0,0) to (3,-2), and the radius stays 16 under the square root, untouched.
This is the exact same move that translates a single point: (x,y) → (x+h, y+k). A circle or ellipse is just a whole cloud of points obeying one rule, so sliding the shape means sliding every point in that cloud the same way, which the algebra does all at once. It connects directly to writing conics in standard form, where h and k are read straight off as the center or vertex, and to transformations of any function, where the same (x-h), (y-k), and leading-coefficient patterns stretch, flip, or relocate graphs beyond conics too.
Key terms
- Center
- The fixed point a circle or ellipse is built around, read directly as (h, k) in standard form.
- Vertex
- The turning point of a parabola, located at (h, k) once the equation is written in shifted form.
- Translation
- A slide of every point of a graph by the same horizontal and vertical amount, with no rotation or resizing.
- Standard form
- A conic's equation written with (x-h) and (y-k) so the center or vertex can be read off directly.
Watch it explained
- How it worksHow do the h and k tucked inside the parentheses relocate a conic without reshaping it?
- Worked exampleFind the center and radius after shifting the circle x2 + y2 = 16 three units right and two down.
- Going furtherWhat happens to an ellipse's equation when its center moves to the point (-2, 5)?
- Ties togetherHow does shifting a whole conic use the very same rule that translates a lone point in the plane?
Check yourself
End of topic.