Free graphing resources · Original practice
Graph four circles from center and radius. Mark the four cardinal points, write standard form and keep coordinate radius separate from printed paper size.
Graph four circles from center and radius. Mark the four cardinal points, write standard form and keep coordinate radius separate from printed paper size.
Choose Letter or A4, then download a student sheet first. Keep the separate reference or answer file closed until you have attempted the case. Every worksheet, SVG and answer is free without signup.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Original worksheet collection. Letter or A4 PDF and editable SVG.
Complete free PDF pack.
| Case | Focus |
|---|---|
| CI01 | Origin circle |
| CI02 | Right-shifted circle |
| CI03 | Upper-left circle |
| CI04 | A larger circle |
All examples use original fictional data and coordinate units. They are practice material, not measured classroom results, published questions or evidence about real people.
CI01: Origin circle
| Check | Worked result |
|---|---|
| 1 | Center (0,0); radius 3. |
| 2 | Cardinal points: (-3,0), (3,0), (0,-3), (0,3). |
| 3 | (x - (0))^2 + (y - (0))^2 = 9. |
The first case shows the checking pattern. Individual answer files contain the completed graph and result for every case.
Begin by identifying the center coordinates provided in the reference PDF for each specific case. For instance, the Origin circle uses center zero zero while the Right-shifted circle uses two negative one. Mark this exact point on your graph paper using a sharp pencil. This step is crucial because the entire circle geometry relies on this fixed anchor point. Do not estimate the location based on visual approximation. Use the grid lines to ensure precision. The center is not the same as the paper diameter. Accurate placement here prevents cumulative errors in subsequent steps. Verify the coordinates against the supplied values before proceeding to the next phase of the construction process.
Once the center is marked, measure the radius distance in four cardinal directions. For the Upper-left circle, move three units up, down, left, and right from center negative two two. Mark these four points clearly. These points define the boundary of the circle. Remember that the radius is a numerical value derived from listed coordinates, not a physical measurement of the printed paper. Use a ruler to ensure straight lines from the center to each cardinal point. This creates a cross shape that guides the final sketch. Precision in these four marks ensures the circle remains symmetrical and true to the mathematical definition provided in the original fictional practice material.
Connect the four cardinal points with a smooth, continuous curve. Avoid jagged lines or sharp corners. The circle should pass evenly through all four marked points. For the larger circle with radius four, ensure the curve extends sufficiently from center one one. Use a light hand to allow for adjustments if necessary. The goal is to represent the equation x minus h squared plus y minus k squared equals r squared visually. Do not rely on the physical diameter of the printed sheet as a guide. The listed coordinates determine the answer. Check your sketch against the answer PDF to verify accuracy. This visual representation completes the geometric construction for each original fictional case.
| Watch for | Check instead |
|---|---|
| Confusing the physical paper diameter with the numerical radius value. | Always use the numerical radius provided in the reference PDF, such as three for the Origin circle, regardless of the printed page size. |
| Estimating the center location without using the grid coordinates. | Plot the exact integer center coordinates, like two negative one for the Right-shifted circle, using the grid lines for precise placement. |
| Drawing a jagged or uneven circle through the cardinal points. | Sketch a smooth curve that passes evenly through the four marked cardinal points to accurately represent the circle definition. |
Choose the PDF that matches your paper. Print at Actual size / 100% and turn off Fit, Shrink and browser headers. SVG is editable but an editor can rescale it; the PDF is the simpler print route. Physical printer, classroom and curriculum tests have not been completed.
Plot the center, then move the radius units left, right, up and down. Use those four points to guide a smooth circle.
Only if physically printed at the intended scale. The exercise radius is defined by coordinate units.
For center (h,k) and radius r, use (x-h)^2+(y-k)^2=r^2.
Yes. The individual student, reference and answer files plus collection packs are free without signup. Credit Zumola under CC BY 4.0 when sharing or adapting.
Take the equation (x - 3)^2 + (y + 2)^2 = 25. Read the center from the subtracted values: x - 3 means the center x-coordinate is 3, and y + 2 means the center y-coordinate is -2, so the center is (3, -2). The right side is r^2, so r = 5 because 5 times 5 is 25. Do not read the radius as 25; that is the squared value.
The four cardinal points sit 5 units from the center along horizontal and vertical lines: (8, -2), (-2, -2), (3, 3) and (3, -7). Plot those four points first, then draw a smooth curve through them. A circle drawn only from a center dot and a guessed sweep usually misses at least one cardinal point, and that miss is the easiest error to catch.
Substitution is the strongest check because it does not depend on your drawing. Put (8, -2) into (x - 3)^2 + (y + 2)^2: (8 - 3)^2 is 25, and (-2 + 2)^2 is 0, so the total is 25, which matches the right side. If a plotted point fails substitution, the point is wrong, not the equation.
Check symmetry next. A circle is symmetric about both the horizontal and vertical lines through its center, so the leftmost and rightmost points should share the same y-coordinate, and the top and bottom points should share the same x-coordinate. If they do not, the center was misread. Finally, check the diameter: the distance between opposite cardinal points should be twice the radius, so here 10 units across, not 5.
A hand-drawn circle has many valid appearances, so judge the construction rather than pixel-perfect roundness. A sketch can be slightly uneven and still be correct if the center, radius and cardinal points are accurate and labeled. Use the rubric below to score a finished attempt, and note which criterion failed rather than marking the whole page wrong.
Keep the coordinate scale explicit. If one grid square represents one unit, the radius of 5 covers five squares in each direction from the center. Mixing scales between axes turns a circle into an oval and makes every later check unreliable.
Graph (x - 3)^2 + (y + 2)^2 = 25 and verify the four cardinal points.
Center (3, -2) and radius 5. Cardinal points: (8, -2), (-2, -2), (3, 3), (3, -7). Check (3, 3): (3 - 3)^2 + (3 + 2)^2 = 0 + 25 = 25, correct. Check (-2, -2): (-2 - 3)^2 + (-2 + 2)^2 = 25 + 0 = 25, correct. Opposite points are 10 units apart, which equals 2r.
Set each bracket to zero: x - 3 = 0 gives x = 3, and y + 2 = 0 gives y = -2, so the center is (3, -2).
Four points fix the center, radius and orientation, and each one can be checked by substitution, which is faster than checking a long table of values.
Recheck the point by substitution first. If the point satisfies the equation, adjust the curve; if it does not, recompute the point from the center and radius.
