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Math Coach Amy
Amy Ferguson Moncure

IB AA · 3B Sketching Graphs

Sketch from Properties

Turn asymptotes and intercepts into one possible graph.

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Key Ideas Easy Medium Hard

Match the breaks, the level line, and the intercepts

What to Remember

Vertical asymptote

The graph breaks at that x-value and runs up or down beside it.

Horizontal asymptote

Far to the left and far to the right, the graph levels off at that y-value.

Zero

The graph crosses the x-axis at the zero.

One possible equation

A fraction with those asymptotes and that zero is enough. The answer shows one equation that fits.

Easy

Easy

The horizontal asymptote is y = 1.

Example
Sketch a graph with these properties: one vertical asymptote, x = 5; a horizontal asymptote, y = 1; one zero, at x = 2; y-intercept (0, 25).

One equation with these properties is y = x − 2x − 5.

xy

Answer: y = x − 2x − 5

1
Sketch a graph with these properties: one vertical asymptote, x = 2; a horizontal asymptote, y = 1; one zero, at x = −3; y-intercept (0, −32).
Show Answer

One equation with these properties is y = x + 3x − 2.

xy

Answer: y = x + 3x − 2

2
Sketch a graph with these properties: one vertical asymptote, x = −1; a horizontal asymptote, y = 1; one zero, at x = 4; y-intercept (0, −4).
Show Answer

One equation with these properties is y = x − 4x + 1.

xy

Answer: y = x − 4x + 1

3
Sketch a graph with these properties: one vertical asymptote, x = 1; a horizontal asymptote, y = 2; one zero, at x = 3; y-intercept (0, 6).
Show Answer

One equation with these properties is y = 2x − 6x − 1.

xy

Answer: y = 2x − 6x − 1

4
Sketch a graph with these properties: one vertical asymptote, x = 3; a horizontal asymptote, y = 1; one zero, at x = −1; y-intercept (0, −13).
Show Answer

One equation with these properties is y = x + 1x − 3.

xy

Answer: y = x + 1x − 3

5
Sketch a graph with these properties: one vertical asymptote, x = −2; a horizontal asymptote, y = 3; one zero, at x = 1; y-intercept (0, −32).
Show Answer

One equation with these properties is y = 3x − 3x + 2.

xy

Answer: y = 3x − 3x + 2

Medium

Medium

The horizontal asymptote is not 1.

Example
Sketch a graph with these properties: one vertical asymptote, x = 4; a horizontal asymptote, y = 1; one zero, at x = 0; y-intercept (0, 0).

One equation with these properties is y = xx − 4.

xy

Answer: y = xx − 4

1
Sketch a graph with these properties: one vertical asymptote, x = 3; a horizontal asymptote, y = 2; one zero, at x = −1; y-intercept (0, −23).
Show Answer

One equation with these properties is y = 2x + 2x − 3.

xy

Answer: y = 2x + 2x − 3

2
Sketch a graph with these properties: one vertical asymptote, x = −2; a horizontal asymptote, y = 1; one zero, at x = 6; y-intercept (0, −3).
Show Answer

One equation with these properties is y = x − 6x + 2.

xy

Answer: y = x − 6x + 2

3
Sketch a graph with these properties: one vertical asymptote, x = 1; a horizontal asymptote, y = 1; one zero, at x = −5; y-intercept (0, −5).
Show Answer

One equation with these properties is y = x + 5x − 1.

xy

Answer: y = x + 5x − 1

4
Sketch a graph with these properties: one vertical asymptote, x = −3; a horizontal asymptote, y = 2; one zero, at x = 2; y-intercept (0, −43).
Show Answer

One equation with these properties is y = 2x − 4x + 3.

xy

Answer: y = 2x − 4x + 3

5
Sketch a graph with these properties: one vertical asymptote, x = 2; a horizontal asymptote, y = 12; one zero, at x = 1; y-intercept (0, 14).
Show Answer

One equation with these properties is y = x − 12x − 4.

xy

Answer: y = x − 12x − 4

Hard

Hard

A fraction coefficient, or a zero at the origin.

Example
Sketch a graph with these properties: one vertical asymptote, x = 2; a horizontal asymptote, y = 3; one zero, at x = −2; y-intercept (0, −3).

One equation with these properties is y = 3x + 6x − 2.

xy

Answer: y = 3x + 6x − 2

1
Sketch a graph with these properties: one vertical asymptote, x = −4; a horizontal asymptote, y = 1; one zero, at x = −2; y-intercept (0, 12).
Show Answer

One equation with these properties is y = x + 2x + 4.

xy

Answer: y = x + 2x + 4

2
Sketch a graph with these properties: one vertical asymptote, x = 5; a horizontal asymptote, y = 2; one zero, at x = 0; y-intercept (0, 0).
Show Answer

One equation with these properties is y = 2xx − 5.

xy

Answer: y = 2xx − 5

3
Sketch a graph with these properties: one vertical asymptote, x = 6; a horizontal asymptote, y = 1; one zero, at x = 3; y-intercept (0, 12).
Show Answer

One equation with these properties is y = x − 3x − 6.

y

Answer: y = x − 3x − 6

4
Sketch a graph with these properties: one vertical asymptote, x = −5; a horizontal asymptote, y = 1; one zero, at x = 8; y-intercept (0, −85).
Show Answer

One equation with these properties is y = x − 8x + 5.

xy

Answer: y = x − 8x + 5

5
Sketch a graph with these properties: one vertical asymptote, x = 3; a horizontal asymptote, y = 2; one zero, at x = −3; y-intercept (0, −2).
Show Answer

One equation with these properties is y = 2x + 6x − 3.

xy

Answer: y = 2x + 6x − 3

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