Problem 33
Question
(a) Write an equation whose graph consists of all points having an abscissa of \(4 .\) (b) Write an equation whose graph consists of all points having an ordinate of \(-3\).
Step-by-Step Solution
Verified Answer
The equations are \( x = 4 \) and \( y = -3 \).
1Step 1: Understand Abscissa and Ordinate
The abscissa represents the x-coordinate of a point, while the ordinate represents the y-coordinate of a point.
2Step 2: Equation for Constant Abscissa
If the abscissa (x-coordinate) of all points is 4, then the x-value is always 4 regardless of the y-value. The equation is: \[ x = 4 \]
3Step 3: Equation for Constant Ordinate
If the ordinate (y-coordinate) of all points is -3, then the y-value is always -3 regardless of the x-value. The equation is: \[ y = -3 \]
Key Concepts
abscissaordinateequations of lines
abscissa
In coordinate geometry, the term 'abscissa' refers to the x-coordinate of a point on the Cartesian plane. It represents the horizontal distance of a point from the y-axis. For instance, in the point (4, -3), the abscissa is 4.
Understanding the abscissa helps in plotting points and creating equations. When given a constant abscissa value, like in the exercise, we are to find the equation representing all such points.
To illustrate, if all points have an abscissa of 4, this means x is always 4, regardless of the y-value. Therefore, the equation of the line is \( x = 4 \). This is a vertical line passing through (4, y) for all y-values.
Understanding the abscissa helps in plotting points and creating equations. When given a constant abscissa value, like in the exercise, we are to find the equation representing all such points.
To illustrate, if all points have an abscissa of 4, this means x is always 4, regardless of the y-value. Therefore, the equation of the line is \( x = 4 \). This is a vertical line passing through (4, y) for all y-values.
ordinate
The ordinate in coordinate geometry is the y-coordinate of a point. It indicates the vertical distance from the x-axis. For the point (4, -3), the ordinate is -3.
Similar to the abscissa, understanding the ordinate allows us to plot points accurately. When the ordinate is fixed, the corresponding x-value can be any number.
Take, for example, the exercise where all points have an ordinate of -3. This tells us the y-value of all points is always -3. Hence, the equation of the line representing these points is \( y = -3 \). This forms a horizontal line intersecting y at -3, covering all x-values.
Similar to the abscissa, understanding the ordinate allows us to plot points accurately. When the ordinate is fixed, the corresponding x-value can be any number.
Take, for example, the exercise where all points have an ordinate of -3. This tells us the y-value of all points is always -3. Hence, the equation of the line representing these points is \( y = -3 \). This forms a horizontal line intersecting y at -3, covering all x-values.
equations of lines
Equations of lines in coordinate geometry describe linear relationships between x and y coordinates. They can be represented in various forms such as slope-intercept form, standard form, and point-slope form.
For vertical lines, where all points share the same x-value, the equation is given as \( x = c \), where c is the constant x-coordinate. For example, \( x = 4 \) represents a vertical line through x=4.
For horizontal lines where all points share the same y-value, the equation is given by \( y = d \), where d is the constant y-coordinate. For instance, \( y = -3 \) represents a horizontal line through y=-3.
Understanding these forms helps in identifying and graphing lines based on their equations, simplifying the concepts of coordinate geometry.
For vertical lines, where all points share the same x-value, the equation is given as \( x = c \), where c is the constant x-coordinate. For example, \( x = 4 \) represents a vertical line through x=4.
For horizontal lines where all points share the same y-value, the equation is given by \( y = d \), where d is the constant y-coordinate. For instance, \( y = -3 \) represents a horizontal line through y=-3.
Understanding these forms helps in identifying and graphing lines based on their equations, simplifying the concepts of coordinate geometry.
Other exercises in this chapter
Problem 32
Prove analytically that the line segments joining consecutive midpoints of the sides of any quadrilateral form a parallelogram.
View solution Problem 33
Prove that if \(a\) and \(b\) are any numbers, then \(|a-b| \leq|a|+|b|\). (HINT: Write \(a-b\) as \(a+(-b)\) and use Theorem 1.2.8.)
View solution Problem 34
Prove that a graph that is symmetric with respect to both coordinate axes is also symmetric with respect to the origin.
View solution Problem 34
Prove analytically that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
View solution