Question

NO LINKS!! Describe the set of all points P(x, y) in a coordinate plane that satisfy the given condition. Part 2​

Answer 1

(d) The set of all points in quadrants I and III (x and y have the same sign).

(e) The set of all points below the x-axis.

(f) The set of all points on the y-axis.

Explanation:

The Cartesian plane is divided into four quadrants.

Quadrant I is the upper right quadrant and the rest follow in a counterclockwise direction.

• Quadrant I: x > 0 and y > 0 → (x, y)
• Quadrant II: x < 0 and y > 0 → (-x, y)
• Quadrant III: x < 0 and y < 0 → (-x, -y)
• Quadrant IV: x > 0 and y < 0 → (x, -y)

Part (d)

For xy > 0, x and y should either both be positive or both be negative, i.e. have the same sign.

• x and y are both positive in quadrant I.

• x and y are both negative in quadrant III.

Therefore, the description that satisfies the given condition is:

• The set of all points in quadrants I and III (x and y have the same sign).

Part (e)

For y < 0, y is always negative.

• y is negative in quadrant III.

• y is negative in quadrant IV.

Quadrants III and IV are below the x-axis.

Therefore, the description that satisfies the given condition is:

• The set of all points below the x-axis.

Part (f)

The only place where x = 0 is the y-axis.

Therefore, the description that satisfies the given condition is:

• The set of all points on the y-axis.

Answer 2

Part (d)

If xy > 0, then the two items x and y must have the same sign.

Examples where x,y are both positive

• x = 2 and y = 5 leads to xy = 2*5 = 10
• x = 7 and y = 4 leads to xy = 7*4 = 28

Examples where x,y are both negative

• x = -1 and y = -9 leads to xy = -1*(-9) = 9
• x = -12 and y = -3 leads to xy = -12*(-3) = 36

This places us in either the first quadrant or third quadrant (Q1 and Q3)

• Q1 is where x > 0 and y > 0 (northeast corner)
• Q3 is where x < 0 and y < 0 (southwest corner)

Answer: The set of all points in quadrants I and III (x and y have the same sign)

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Part (e)

If y < 0, then we have points like (2,-5) and (1,-7). The x coordinate doesn't matter as long as the y coordinate is negative.

Visually speaking we are below the x axis. This places the point in Q3, Q4, or on the negative y axis. The point cannot be on the x axis. You can think of this point as being underground or underwater.

Answer: The set of all points below the x axis

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Part (f)

If x = 0, then the point is on the vertical y axis. Recall that y intercepts always occur when x = 0. Two examples would be (0,4) and (0,12).

Answer: The set of all points on the y axis.