Graphs in Context | Math Coach Amy

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Key Ideas Easy Medium Hard

Match the story to a line, a parabola, or a level asymptote

What to Remember

A maximum profit

For a quadratic P = aq² + bq + c with a < 0, the maximum is at q = −b2a.

An exponential level

y = A − B e^{kx} levels off at y = A when the exponential fades.

Equal steps

If the outputs drop by the same amount each second, a straight line fits.

Asymptotes in a formula

Denominator 0 gives a vertical asymptote. A constant numerator over x² levels off at y = 0.

Easy

A line, a profit vertex, or one rational graph.

Example

A workshop's monthly profit, in dollars, is P = 9q − q²40 − 180, where q is the number of items. How many items give the maximum profit, and what is that profit?

P = −140q² + 9q − 180

q = −b2a

a = −140 and b = 9

q = −92 · (−140)

q = 9120

q = 180

P = 9(180) − 32400 ÷ 40 − 180

P = 1620 − 810 − 180

P = 630

Answer: 180 items, profit $630

Sketch y = 8 − 12e^−0.4x. State the y-intercept, the horizontal asymptote, and the x-intercept to 3 significant figures.

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y = 8 − 12e^−0.4x

At x = 0, y = 8 − 12 = −4. The y-intercept is (0, −4).

As x grows, the exponential term fades, so the horizontal asymptote is y = 8.

The graph crosses the x-axis when 12e^−0.4x = 8.

x = 1.01

Answer: y-intercept (0, −4), HA y = 8, x-intercept about (1.01, 0)

A car is moving at 24 m/s when the driver brakes. After 1 second the speed is 16 m/s, and after 2 seconds it is 8 m/s. Sketch a possible graph of speed against time for these three moments, and write a possible equation.

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The drops are equal, so a straight line fits the three points.

v = 24 − 8t

The graph passes through (0, 24), (1, 16), and (2, 8).

Answer: v = 24 − 8t

A shop loses $80 in a month with no advertising. Profit is $130 when they spend $50 or $350. The maximum profit is $400 when they spend $200. Sketch a possible graph of profit against advertising spend.

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A parabola with vertex (200, 400) matches a single maximum.

It also passes through (0, −80), (50, 130), and (350, 130).

One equation is P = −3250(q − 200)² + 400.

Answer: a parabola through (0, −80) with vertex (200, 400)

Sketch y = 1x² − 16. State the equations of the asymptotes.

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The denominator is 0 when x² = 16, so x = 4 or x = −4.

The vertical asymptotes are x = −4 and x = 4.

For large x the fraction is near 0, so the horizontal asymptote is y = 0.

There is no x-intercept, because the numerator is 1.

The y-intercept is (0, −116).

Answer: VA x = −4 and x = 4, HA y = 0

Sketch y = ln(x² − 4x + 13) and give the coordinates of the vertex.

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(x − 2)² + 9 is smallest at x = 2, with value 9.

The vertex is (2, ln(9)).

Answer: (2, ln(9))

Medium

A situation to sketch.

Example

Find the line of symmetry of y = 1x² + 4x + 7.

x² + 4x + 7 = (x + 2)² + 3

The denominator, and so the graph, is symmetric about x = −2.

Answer: x = −2

A test was marked out of 50. The lowest mark was 10 and the highest was 48. Exactly half of the 40 students scored under 30. Sketch a possible graph of N, the number of students who scored fewer than m marks.

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N starts at 0 when m is 10, because nobody scored under 10.

N is 20 when m is 30.

N is 40 when m reaches 48.

The graph rises from (10, 0) through (30, 20) to (48, 40).

Answer: a rising graph through (10, 0), (30, 20), and (48, 40)

For y = 2x − 1x + 3, state the domain and the equations of any asymptotes.

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y = 2x − 1x + 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 3 = 0

x = −3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x − 1 = 0

x = 12

The x-intercept is (12, 0).

At x = 0,

y = −13

The y-intercept is (0, −13).

Answer: VA x = −3, HA y = 2, x-intercept (12, 0), y-intercept (0, −13)

Sketch a graph with a vertical asymptote x = 1, a horizontal asymptote y = 3, and only the intercepts (−1, 0) and (0, −3).

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One equation is y = 3x + 3x − 1.

Check: zero when 3x + 3 = 0, so x = −1. At x = 0, y = −3.

Answer: y = 3x + 3x − 1

How many items give the maximum of P = 6q − 0.02q² − 120?

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q = 60.04

q = 150

Answer: 150 items

For y = x − 3x + 2, write the equations of the asymptotes and the intercepts.

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y = x − 3x + 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 2 = 0

x = −2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x − 3 = 0

x = 3

The x-intercept is (3, 0).

At x = 0,

y = −32

The y-intercept is (0, −32).

Answer: VA x = −2, HA y = 1, x-intercept (3, 0), y-intercept (0, −32)

Hard

Asymptotes together with a model.

Example

For y = 5x − 1, write the equations of the asymptotes and the intercepts.

y = 5x − 1

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 1 = 0

x = 1

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 0

The numerator is never 0, so there is no x-intercept.

At x = 0,

y = −5

The y-intercept is (0, −5).

Answer: VA x = 1, HA y = 0, no x-intercept, y-intercept (0, −5)

For y = 2xx − 4, write the equations of the asymptotes and the intercepts.

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y = 2xx − 4

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 4 = 0

x = 4

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x = 0

x = 0

The x-intercept is (0, 0).

At x = 0,

y = 0

The y-intercept is (0, 0).

Answer: VA x = 4, HA y = 2, x-intercept (0, 0), y-intercept (0, 0)

For y = x + 4x − 2, write the equations of the asymptotes and the intercepts.

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y = x + 4x − 2

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 2 = 0

x = 2

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x + 4 = 0

x = −4

The x-intercept is (−4, 0).

At x = 0,

y = −2

The y-intercept is (0, −2).

Answer: VA x = 2, HA y = 1, x-intercept (−4, 0), y-intercept (0, −2)

For y = 3x − 9x + 3, write the equations of the asymptotes and the intercepts.

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y = 3x − 9x + 3

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 3 = 0

x = −3

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 3

3x − 9 = 0

x = 3

The x-intercept is (3, 0).

At x = 0,

y = −3

The y-intercept is (0, −3).

Answer: VA x = −3, HA y = 3, x-intercept (3, 0), y-intercept (0, −3)

For y = x − 5x + 6, write the equations of the asymptotes and the intercepts.

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y = x − 5x + 6

A vertical asymptote is where the denominator is 0 and the numerator is not.

x + 6 = 0

x = −6

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 1

x − 5 = 0

x = 5

The x-intercept is (5, 0).

At x = 0,

y = −56

The y-intercept is (0, −56).

Answer: VA x = −6, HA y = 1, x-intercept (5, 0), y-intercept (0, −56)

For y = 2x + 8x − 4, write the equations of the asymptotes and the intercepts.

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y = 2x + 8x − 4

A vertical asymptote is where the denominator is 0 and the numerator is not.

x − 4 = 0

x = 4

For a linear fraction, the horizontal asymptote is the ratio of the x coefficients.

y = 2

2x + 8 = 0

x = −4

The x-intercept is (−4, 0).

At x = 0,

y = −2

The y-intercept is (0, −2).

Answer: VA x = 4, HA y = 2, x-intercept (−4, 0), y-intercept (0, −2)

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