Points of Intersection | Math Coach Amy

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Key Ideas Easy Medium Hard

Set the y-expressions equal

What to Remember

One equation

At a shared point the two expressions for y are equal.

Then factor

Bring every term to one side. Factor, or use a method you already trust.

Both coordinates

Each x gives a y. Write the point (x, y).

A decimal root

If the equation does not factor, a graphing calculator can give the x-coordinate to 3 significant figures. Substitute to find y.

Easy

The equation factors.

Example

Find the points where y = x² − 5 meets y = x + 1.

Set the expressions equal.

x² − 5 = x + 1

x² − x − 6 = 0

(x − 3)(x + 2) = 0

x = 3 or x = −2

y = 4 when x = 3.

y = −1 when x = −2.

Answer: (3, 4), (−2, −1)

Find the points where y = x² − 4 meets y = 5.

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Set the expressions equal.

x² − 4 = 5

x² = 9

x = 3 or x = −3

y = 5 at both points.

Answer: (3, 5), (−3, 5)

Find the points where y = 2x + 1 meets y = −x + 7.

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Set the expressions equal.

2x + 1 = −x + 7

3x = 6

x = 2

y = 5

Answer: (2, 5)

Find the points where y = x² − 1 meets y = 3x − 3.

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Set the expressions equal.

x² − 1 = 3x − 3

x² − 3x + 2 = 0

(x − 1)(x − 2) = 0

x = 1 or x = 2

The points are (1, 0) and (2, 3).

Answer: (1, 0), (2, 3)

Find the points where y = x² meets y = 4.

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Set the expressions equal.

x² = 4

x = 2 or x = −2

y = 4

Answer: (2, 4), (−2, 4)

Find the points where y = x² − 2x meets y = 3x − 6.

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Set the expressions equal.

x² − 2x = 3x − 6

x² − 5x + 6 = 0

(x − 2)(x − 3) = 0

x = 2 or x = 3

The points are (2, 0) and (3, 3).

Answer: (2, 0), (3, 3)

Medium

A cubic or two quadratics.

Example

Find the points where y = x³ − x meets y = 0.

Set the expressions equal.

x³ − x = 0

x(x² − 1) = 0

x(x − 1)(x + 1) = 0

x = −1, 0, or 1

Every y-coordinate is 0.

Answer: (−1, 0), (0, 0), (1, 0)

Find the points where y = x² + 3x meets y = 4x + 6.

Show Answer

Set the expressions equal.

x² + 3x = 4x + 6

x² − x − 6 = 0

(x − 3)(x + 2) = 0

x = 3 or x = −2

The points are (3, 18) and (−2, −2).

Answer: (3, 18), (−2, −2)

Find the points where y = x³ meets y = x.

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Set the expressions equal.

x³ = x

x³ − x = 0

x(x − 1)(x + 1) = 0

x = −1, 0, or 1

Answer: (−1, −1), (0, 0), (1, 1)

Find the points where y = x² − 6x + 5 meets y = −3.

Show Answer

Set the expressions equal.

x² − 6x + 5 = −3

x² − 6x + 8 = 0

(x − 2)(x − 4) = 0

y = −3 at both points.

Answer: (2, −3), (4, −3)

Find the points where y = x² + x − 2 meets y = 2x − 2.

Show Answer

Set the expressions equal.

x² + x − 2 = 2x − 2

x² − x = 0

x(x − 1) = 0

x = 0 or x = 1

The points are (0, −2) and (1, 0).

Answer: (0, −2), (1, 0)

Find the points where y = 2x − 4 meets y = −3x + 6.

Show Answer

Set the expressions equal.

2x − 4 = −3x + 6

5x = 10

x = 2

y = 0

Answer: (2, 0)

Hard

A root that needs 3 significant figures.

Example

Find the intersection of y = 5 − x and y = 12e^x, to 3 significant figures.

5 − x = 12e^x

This equation does not factor. Graph both sides, or graph their difference.

A graphing calculator gives x = 1.84.

y = 5 − x = 3.16.

Answer: (1.84, 3.16)

Find the positive root of x² + 2 = eˣ, to 3 significant figures. That root is an x-coordinate where y = x² + 2 meets y = eˣ.

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x² + 2 = eˣ

Graph y = x² + 2 − eˣ and read the positive x-intercept.

x = 1.32

Answer: 1.32

Find the real root of x³ + x − 6 = 0 to 3 significant figures.

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x³ + x − 6 = 0

The graph of y = x³ + x − 6 crosses the x-axis once.

x = 1.63

Answer: 1.63

Find the intersection of y = 4 − x and y = 3e^−0.3x that lies to the right of the y-axis. Give both coordinates to 3 significant figures.

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4 − x = 3e^−0.3x

x = 2.64

y = 1.36

Answer: (2.64, 1.36)

Find the points where y = x − 2 meets y = −2x + 7.

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Set the expressions equal.

x − 2 = −2x + 7

3x = 9

x = 3

y = 1

Answer: (3, 1)

Find the points where y = x² − 9 meets y = 0.

Show Answer

Set the expressions equal.

x² − 9 = 0

(x − 3)(x + 3) = 0

The points are (3, 0) and (−3, 0).

Answer: (3, 0), (−3, 0)

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