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Conics and rational graph features

Derive coordinate restrictions, distinguish holes from asymptotes, and use equations to verify graph features.

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Take a moment to read at your own pace. Choose the text size and spacing that feel comfortable for you.

Pause after each idea, then explain it in your own words before moving on.

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Enlarged diagram

Let the equation decide the allowed coordinates

A graph is the set of coordinate pairs satisfying its equation. It need not be a single-valued function of x. Before choosing a graphing window, identify restrictions from squares, denominators and the stated domain. These restrictions can reveal a missing branch or an impossible point before you plot anything.

For y² = ax with a > 0, the left side is nonnegative, so x ≥ 0. The vertex is (0,0) and the parabola opens right. Every positive x gives two y-values, y = ± square root of ax, while every real y gives an allowed x. The whole curve has range R and is symmetric about y = 0; it is not one function y = f(x).

With a < 0, the allowed inputs instead satisfy x ≤ 0, and the parabola opens left. The parameter’s sign changes the opening, not the nonnegativity of a square.

For x² = by with b > 0, the restriction is y ≥ 0. This parabola opens upward, has vertex (0,0) and symmetry axis x = 0, and is a function y = x²/b on all real x. A negative b reverses the opening and gives y ≤ 0. The cases a = 0 or b = 0 reduce to straight lines; they are not parabolas.

Check your understanding 1 A sideways curve has two branches

For y² = -4x, state the possible x- and y-values, opening direction and symmetry axis. Find all points on the curve with x = -1. Is the whole curve one function of x?

Show answer

Require -4x ≥ 0, so x ≤ 0; every real y is possible because x = -y²/4. The parabola opens left and is symmetric about y = 0. At x = -1, y² = 4, giving both (-1,2) and (-1,-2). A vertical line through x = -1 meets it twice, so the whole relation is not a single-valued function of x.

An ellipse is bounded in both coordinates

For positive a and b,

x²/a² + y²/b² = 1.

Both terms are nonnegative and sum to one. Therefore |x| ≤ a and |y| ≤ b. Setting one coordinate to zero gives intercepts (± a,0) and (0,± b). Changing the sign of x or y leaves the equation unchanged, so both coordinate axes are symmetry axes. The ellipse is closed and has no asymptotes.

For example, x²/16 + y²/9 = 1 extends from -4 to 4 horizontally and from -3 to 3 vertically. At x = 0 it contains both (0,3) and (0,-3). The upper half alone is the function y = 3 square root of (1-x²/16) on [-4,4]; its range is [0,3]. Selecting that branch changes the relation you are describing.

The complete ellipse x²/16 + y²/9 = 1

A closed ellipse, wider horizontally than vertically, meets the axes at (−4, 0), (4, 0), (0, −3) and (0, 3). Both coordinate axes are symmetry axes. Equal coordinate scales preserve its shape.

Scroll across the figure to read all labels.

A closed ellipse, wider horizontally than vertically, meets the axes at (−4, 0), (4, 0), (0, −3) and (0, 3). Both coordinate axes are symmetry axes. Equal coordinate scales preserve its shape.A closed ellipse, wider horizontally than vertically, meets the axes at (−4, 0), (4, 0), (0, −3) and (0, 3). Both coordinate axes are symmetry axes. Equal coordinate scales preserve its shape.
Filled squares mark the four included axis intercepts. The whole ellipse has −4 ≤ x ≤ 4 and −3 ≤ y ≤ 3; the upper half alone would omit the negative y-values.
Open full-size graphOpen full-size graph
What to noticeClosely spaced samples guide the smooth curve; displayed table values are rounded. The entire ellipse lies inside this window.
View figure data
Figure data: The complete ellipse x²/16 + y²/9 = 1
Seriesx (dimensionless)y (dimensionless)
Complete ellipse4.00000.0000
Complete ellipse3.99140.1962
Complete ellipse3.96580.3916
Complete ellipse3.92310.5853
Complete ellipse3.86370.7765
Complete ellipse3.78770.9643
Complete ellipse3.69551.1481
Complete ellipse3.58751.3269
Complete ellipse3.46411.5000
Complete ellipse3.32591.6667
Complete ellipse3.17341.8263
Complete ellipse3.00741.9780
Complete ellipse2.82842.1213
Complete ellipse2.63742.2555
Complete ellipse2.43502.3801
Complete ellipse2.22232.4944
Complete ellipse2.00002.5981
Complete ellipse1.76922.6906
Complete ellipse1.53072.7716
Complete ellipse1.28582.8408
Complete ellipse1.03532.8978
Complete ellipse0.78042.9424
Complete ellipse0.52212.9743
Complete ellipse0.26162.9936
Complete ellipse0.00003.0000
Complete ellipse-0.26162.9936
Complete ellipse-0.52212.9743
Complete ellipse-0.78042.9424
Complete ellipse-1.03532.8978
Complete ellipse-1.28582.8408
Complete ellipse-1.53072.7716
Complete ellipse-1.76922.6906
Complete ellipse-2.00002.5981
Complete ellipse-2.22232.4944
Complete ellipse-2.43502.3801
Complete ellipse-2.63742.2555
Complete ellipse-2.82842.1213
Complete ellipse-3.00741.9780
Complete ellipse-3.17341.8263
Complete ellipse-3.32591.6667
Complete ellipse-3.46411.5000
Complete ellipse-3.58751.3269
Complete ellipse-3.69551.1481
Complete ellipse-3.78770.9643
Complete ellipse-3.86370.7765
Complete ellipse-3.92310.5853
Complete ellipse-3.96580.3916
Complete ellipse-3.99140.1962
Complete ellipse-4.00000.0000
Complete ellipse-3.9914-0.1962
Complete ellipse-3.9658-0.3916
Complete ellipse-3.9231-0.5853
Complete ellipse-3.8637-0.7765
Complete ellipse-3.7877-0.9643
Complete ellipse-3.6955-1.1481
Complete ellipse-3.5875-1.3269
Complete ellipse-3.4641-1.5000
Complete ellipse-3.3259-1.6667
Complete ellipse-3.1734-1.8263
Complete ellipse-3.0074-1.9780
Complete ellipse-2.8284-2.1213
Complete ellipse-2.6374-2.2555
Complete ellipse-2.4350-2.3801
Complete ellipse-2.2223-2.4944
Complete ellipse-2.0000-2.5981
Complete ellipse-1.7692-2.6906
Complete ellipse-1.5307-2.7716
Complete ellipse-1.2858-2.8408
Complete ellipse-1.0353-2.8978
Complete ellipse-0.7804-2.9424
Complete ellipse-0.5221-2.9743
Complete ellipse-0.2616-2.9936
Complete ellipse0.0000-3.0000
Complete ellipse0.2616-2.9936
Complete ellipse0.5221-2.9743
Complete ellipse0.7804-2.9424
Complete ellipse1.0353-2.8978
Complete ellipse1.2858-2.8408
Complete ellipse1.5307-2.7716
Complete ellipse1.7692-2.6906
Complete ellipse2.0000-2.5981
Complete ellipse2.2223-2.4944
Complete ellipse2.4350-2.3801
Complete ellipse2.6374-2.2555
Complete ellipse2.8284-2.1213
Complete ellipse3.0074-1.9780
Complete ellipse3.1734-1.8263
Complete ellipse3.3259-1.6667
Complete ellipse3.4641-1.5000
Complete ellipse3.5875-1.3269
Complete ellipse3.6955-1.1481
Complete ellipse3.7877-0.9643
Complete ellipse3.8637-0.7765
Complete ellipse3.9231-0.5853
Complete ellipse3.9658-0.3916
Complete ellipse3.9914-0.1962
Complete ellipse4.00000.0000
Included intercepts-4.00000.0000
Included intercepts4.00000.0000
Included intercepts0.0000-3.0000
Included intercepts0.00003.0000

On a calculator that accepts only y as a function of x, enter both Y₁ = 3 square root of (1-X²/16) and Y₂ = -3 square root of (1-X²/16). Their real inputs stop at X = ±4. A window such as -5 ≤ X ≤ 5, -4 ≤ Y ≤ 4 contains the whole ellipse; software supporting implicit equations can instead plot the original relation directly.

The denominators are squared intercept lengths. Reading 16 as the horizontal intercept or drawing only the positive square-root branch would give a different graph.

The positive squared term locates hyperbola branches

For the hyperbola

x²/a²-y²/b² = 1, a,b > 0,

rearrange to x²/a² = 1 + y²/b² ≥ 1. Thus x ≤ -a or x ≥ a. Its branches open left and right, with vertices (± a,0). Every real y is possible. Both coordinate axes are symmetry axes, but neither branch enters the strip -a < x < a.

For

y²/b²-x²/a² = 1,

instead y²/b² = 1 + x²/a² ≥ 1. The branches open up and down, with vertices (0,± b), all real x allowed, and y ≤ -b or y ≥ b. The positive squared term identifies the opening direction.

For either orientation, the asymptotes are y = ±(b/a)x. Far from the origin, the constant one is small relative to the squared terms, suggesting y²/b² ≈ x²/a². These lines describe limiting behaviour; they are not parts of the hyperbola. Substituting either line into its equation gives zero on the left, not one.

A hyperbola opening up and down

The two branches have vertices (0, 3) and (0, −3), with no curve in −3 < y < 3. Every real x is allowed. Both coordinate axes are symmetry axes. Asymptotes y = ±3x/4 approach the branches far from the origin.

Scroll across the figure to read all labels.

The two branches have vertices (0, 3) and (0, −3), with no curve in −3 < y < 3. Every real x is allowed. Both coordinate axes are symmetry axes. Asymptotes y = ±3x/4 approach the branches far from the origin.The two branches have vertices (0, 3) and (0, −3), with no curve in −3 < y < 3. Every real x is allowed. Both coordinate axes are symmetry axes. Asymptotes y = ±3x/4 approach the branches far from the origin.
The solid upper and dashed lower branches satisfy y²/9 − x²/16 = 1. The dotted and dash-dot diagonal lines are asymptotes; they are not parts of the hyperbola.
Open full-size graphOpen full-size graph
What to noticeEqual coordinate scales show the geometry. Both branches continue outside this window. Sample coordinates and displayed table values are approximate; use the equation for exact features.
View figure data
Figure data: A hyperbola opening up and down
Seriesx (dimensionless)y (dimensionless)
Upper branch: y ≥ 3-6.005.408
Upper branch: y ≥ 3-5.855.315
Upper branch: y ≥ 3-5.705.223
Upper branch: y ≥ 3-5.555.131
Upper branch: y ≥ 3-5.405.040
Upper branch: y ≥ 3-5.254.950
Upper branch: y ≥ 3-5.104.861
Upper branch: y ≥ 3-4.954.773
Upper branch: y ≥ 3-4.804.686
Upper branch: y ≥ 3-4.654.600
Upper branch: y ≥ 3-4.504.516
Upper branch: y ≥ 3-4.354.432
Upper branch: y ≥ 3-4.204.350
Upper branch: y ≥ 3-4.054.269
Upper branch: y ≥ 3-3.904.190
Upper branch: y ≥ 3-3.754.112
Upper branch: y ≥ 3-3.604.036
Upper branch: y ≥ 3-3.453.962
Upper branch: y ≥ 3-3.303.889
Upper branch: y ≥ 3-3.153.819
Upper branch: y ≥ 3-3.003.750
Upper branch: y ≥ 3-2.853.684
Upper branch: y ≥ 3-2.703.619
Upper branch: y ≥ 3-2.553.558
Upper branch: y ≥ 3-2.403.499
Upper branch: y ≥ 3-2.253.442
Upper branch: y ≥ 3-2.103.388
Upper branch: y ≥ 3-1.953.338
Upper branch: y ≥ 3-1.803.290
Upper branch: y ≥ 3-1.653.245
Upper branch: y ≥ 3-1.503.204
Upper branch: y ≥ 3-1.353.166
Upper branch: y ≥ 3-1.203.132
Upper branch: y ≥ 3-1.053.102
Upper branch: y ≥ 3-0.903.075
Upper branch: y ≥ 3-0.753.052
Upper branch: y ≥ 3-0.603.034
Upper branch: y ≥ 3-0.453.019
Upper branch: y ≥ 3-0.303.008
Upper branch: y ≥ 3-0.153.002
Upper branch: y ≥ 30.003.000
Upper branch: y ≥ 30.153.002
Upper branch: y ≥ 30.303.008
Upper branch: y ≥ 30.453.019
Upper branch: y ≥ 30.603.034
Upper branch: y ≥ 30.753.052
Upper branch: y ≥ 30.903.075
Upper branch: y ≥ 31.053.102
Upper branch: y ≥ 31.203.132
Upper branch: y ≥ 31.353.166
Upper branch: y ≥ 31.503.204
Upper branch: y ≥ 31.653.245
Upper branch: y ≥ 31.803.290
Upper branch: y ≥ 31.953.338
Upper branch: y ≥ 32.103.388
Upper branch: y ≥ 32.253.442
Upper branch: y ≥ 32.403.499
Upper branch: y ≥ 32.553.558
Upper branch: y ≥ 32.703.619
Upper branch: y ≥ 32.853.684
Upper branch: y ≥ 33.003.750
Upper branch: y ≥ 33.153.819
Upper branch: y ≥ 33.303.889
Upper branch: y ≥ 33.453.962
Upper branch: y ≥ 33.604.036
Upper branch: y ≥ 33.754.112
Upper branch: y ≥ 33.904.190
Upper branch: y ≥ 34.054.269
Upper branch: y ≥ 34.204.350
Upper branch: y ≥ 34.354.432
Upper branch: y ≥ 34.504.516
Upper branch: y ≥ 34.654.600
Upper branch: y ≥ 34.804.686
Upper branch: y ≥ 34.954.773
Upper branch: y ≥ 35.104.861
Upper branch: y ≥ 35.254.950
Upper branch: y ≥ 35.405.040
Upper branch: y ≥ 35.555.131
Upper branch: y ≥ 35.705.223
Upper branch: y ≥ 35.855.315
Upper branch: y ≥ 36.005.408
Lower branch: y ≤ −3-6.00-5.408
Lower branch: y ≤ −3-5.85-5.315
Lower branch: y ≤ −3-5.70-5.223
Lower branch: y ≤ −3-5.55-5.131
Lower branch: y ≤ −3-5.40-5.040
Lower branch: y ≤ −3-5.25-4.950
Lower branch: y ≤ −3-5.10-4.861
Lower branch: y ≤ −3-4.95-4.773
Lower branch: y ≤ −3-4.80-4.686
Lower branch: y ≤ −3-4.65-4.600
Lower branch: y ≤ −3-4.50-4.516
Lower branch: y ≤ −3-4.35-4.432
Lower branch: y ≤ −3-4.20-4.350
Lower branch: y ≤ −3-4.05-4.269
Lower branch: y ≤ −3-3.90-4.190
Lower branch: y ≤ −3-3.75-4.112
Lower branch: y ≤ −3-3.60-4.036
Lower branch: y ≤ −3-3.45-3.962
Lower branch: y ≤ −3-3.30-3.889
Lower branch: y ≤ −3-3.15-3.819
Lower branch: y ≤ −3-3.00-3.750
Lower branch: y ≤ −3-2.85-3.684
Lower branch: y ≤ −3-2.70-3.619
Lower branch: y ≤ −3-2.55-3.558
Lower branch: y ≤ −3-2.40-3.499
Lower branch: y ≤ −3-2.25-3.442
Lower branch: y ≤ −3-2.10-3.388
Lower branch: y ≤ −3-1.95-3.338
Lower branch: y ≤ −3-1.80-3.290
Lower branch: y ≤ −3-1.65-3.245
Lower branch: y ≤ −3-1.50-3.204
Lower branch: y ≤ −3-1.35-3.166
Lower branch: y ≤ −3-1.20-3.132
Lower branch: y ≤ −3-1.05-3.102
Lower branch: y ≤ −3-0.90-3.075
Lower branch: y ≤ −3-0.75-3.052
Lower branch: y ≤ −3-0.60-3.034
Lower branch: y ≤ −3-0.45-3.019
Lower branch: y ≤ −3-0.30-3.008
Lower branch: y ≤ −3-0.15-3.002
Lower branch: y ≤ −30.00-3.000
Lower branch: y ≤ −30.15-3.002
Lower branch: y ≤ −30.30-3.008
Lower branch: y ≤ −30.45-3.019
Lower branch: y ≤ −30.60-3.034
Lower branch: y ≤ −30.75-3.052
Lower branch: y ≤ −30.90-3.075
Lower branch: y ≤ −31.05-3.102
Lower branch: y ≤ −31.20-3.132
Lower branch: y ≤ −31.35-3.166
Lower branch: y ≤ −31.50-3.204
Lower branch: y ≤ −31.65-3.245
Lower branch: y ≤ −31.80-3.290
Lower branch: y ≤ −31.95-3.338
Lower branch: y ≤ −32.10-3.388
Lower branch: y ≤ −32.25-3.442
Lower branch: y ≤ −32.40-3.499
Lower branch: y ≤ −32.55-3.558
Lower branch: y ≤ −32.70-3.619
Lower branch: y ≤ −32.85-3.684
Lower branch: y ≤ −33.00-3.750
Lower branch: y ≤ −33.15-3.819
Lower branch: y ≤ −33.30-3.889
Lower branch: y ≤ −33.45-3.962
Lower branch: y ≤ −33.60-4.036
Lower branch: y ≤ −33.75-4.112
Lower branch: y ≤ −33.90-4.190
Lower branch: y ≤ −34.05-4.269
Lower branch: y ≤ −34.20-4.350
Lower branch: y ≤ −34.35-4.432
Lower branch: y ≤ −34.50-4.516
Lower branch: y ≤ −34.65-4.600
Lower branch: y ≤ −34.80-4.686
Lower branch: y ≤ −34.95-4.773
Lower branch: y ≤ −35.10-4.861
Lower branch: y ≤ −35.25-4.950
Lower branch: y ≤ −35.40-5.040
Lower branch: y ≤ −35.55-5.131
Lower branch: y ≤ −35.70-5.223
Lower branch: y ≤ −35.85-5.315
Lower branch: y ≤ −36.00-5.408
Asymptote y = 3x/4-6.00-4.500
Asymptote y = 3x/46.004.500
Asymptote y = −3x/4-6.004.500
Asymptote y = −3x/46.00-4.500

Check your understanding 2 Read the positive squared term

Give the vertices, coordinate restrictions, symmetry axes and asymptotes of y²/9-x²/16 = 1. Explain why x = ±4 are not its vertices.

Show answer

The branches open up and down, with vertices (0,3) and (0,-3). All real x are allowed, while y ≤ -3 or y ≥ 3. Its symmetry axes are x = 0 and y = 0, and the asymptotes are y = ±3x/4.

Setting y = 0 would require -x²/16 = 1, which has no real solution. Thus there are no x-intercepts; the number 4 determines the asymptote slopes together with 3, not an x-intercept of this orientation.

A rational graph’s exclusions survive rearrangement

For

y = (2x + 1)/(x-3) = 2 + 7/(x-3),

the original denominator excludes x = 3. The final term is unbounded as x approaches 3, giving a vertical asymptote there. It tends to zero as |x| grows, giving horizontal asymptote y = 2. This graph never attains y = 2, since 7/(x-3) cannot equal zero. Its intercepts are (-1/2,0) and (0,-1/3).

Write u = x-3 and v = y-2. The equation becomes uv = 7. Replacing (u,v) by (-u,-v) preserves it, so the graph has centre of rotational symmetry (3,2). Swapping u and v also preserves it; its two reflection axes are y-2 = ±(x-3). The vertical and horizontal asymptotes are not those reflection axes. The centre itself is excluded from the curve.

A denominator zero does not always produce an asymptote. For

y = (x²-1)/(x-1), x ≠ 1,

cancelling the common factor gives y = x + 1 only at allowed inputs. The graph is that line with a hole at (1,2). Nearby outputs tend to 2, rather than becoming unbounded. Its range excludes 2 because the only input producing that output on the line was removed.

Check your understanding 3 Decide what happens at the removed input

A plotter simplifies y = (x²-4)/(x-2) and draws a complete straight line. Give the original domain, the missing point and the range. Is x = 2 a vertical asymptote?

Show answer

The original domain is Rsetminus{2}. At allowed inputs, y = x + 2, so the graph has a hole at (2,4) and range Rsetminus{4}. As x approaches 2, y tends to 4; there is no vertical asymptote there. The simplified expression alone restores a point that the original equation excludes.

Division reveals a sloping asymptote and exact turning levels

For a quadratic numerator and linear denominator, divide before interpreting the graph:

y = (x² + 1)/(x-1) = x + 1 + 2/(x-1), x ≠ 1.

The vertical asymptote is x = 1. The difference between the curve and the line y = x + 1 is 2/(x-1), which tends to zero as |x| grows. Therefore the line is the oblique asymptote. A ratio close to one or a line that merely looks close in a narrow window is not the defining check.

There is no x-intercept because x² + 1 is always positive; the y-intercept is (0,-1). To find which output levels occur without relying on a picture, put u = x-1. Then

y-2 = u + 2/u, u²-(y-2)u + 2 = 0.

For a real point at a given output y, this quadratic in u needs discriminant (y-2)²-8 ≥ 0. Its roots cannot be zero, since their product is 2. Hence the exact range is

y ≤ 2-2 square root of 2 or y ≥ 2 + 2 square root of 2.

At either boundary the quadratic has a repeated root, u = ± square root of 2. This gives turning points (1- square root of 2,2-2 square root of 2) and (1 + square root of 2,2 + 2 square root of 2): a maximum on the left branch and a minimum on the right. The range gap also confirms that an apparent x-axis crossing on a coarse plot is false. Later, differentiation gives another way to locate the same turning points.

Try it yourself 1

Check the width of an opening

Minimal support

Problem

An ideal opening has upper boundary x²/16 + y²/9 = 1 with y ≥ 0 and a level floor y = 0. The numerical coordinates x and y are measured in metres. Model an object as a rectangle centred at x = 0, standing on the floor, with height 2.4 m. Does width 5 m fit without crossing the boundary? Find the greatest width if contact with the boundary is allowed, then state how the answer changes when positive clearance is required. Explain which part of the rectangle you check and why.

Write your answer, then compare it with the solution

Hints

Hint 1

Compare the rectangle with the allowed upper branch

The boundary height changes with horizontal distance from the centre. Relate the rectangle’s width to the coordinates of its upper corners before testing whether it fits.

Show full solution

  1. Locate the limiting corners

    Method

    Use the upper branch and the rectangle’s half-width.

    Reason

    The height 3 square root of (1-x²/16) decreases as |x| increases within the opening. The upper corners therefore face the smallest boundary height above the rectangle.

    Working

    Answer

    For width 5, the upper corners have coordinates (±2.5,2.4). Their substitution gives 2.5²/16 + 2.4²/9 = 0.390625 + 0.64 = 1.030625 > 1. They lie outside the ellipse, so the modelled rectangle does not fit.

  2. Separate contact from positive clearance

    Method

    Replace the width by a nonnegative variable w and solve the boundary condition.

    Reason

    Equality represents contact at the upper corners. Positive clearance requires a strict inequality.

    Working

    Answer

    Require (w/2)²/16 + 0.64 ≤ 1, so w² ≤ 23.04 and 0 ≤ w ≤ 4.8. The greatest width with contact allowed is 4.8 m. Positive clearance requires w < 4.8 m; 4.8 is then a limiting width, not an attained maximum.

For a complete sketch, record the original domain, coordinate restrictions, intercepts, symmetry and asymptotes before accepting a software plot. Use separate function branches when a relation has two y-values, and check excluded inputs algebraically. An empty part of one display window does not prove that a branch or intercept does not exist.

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