Matching Equations with Their Graphs (3.12.3)

Question 1

• Each type of equation has its own signature graph. Learn to match graphs with their equations.

Answer 1

• Each type of equation has its own signature graph. Learn to match graphs with their equations.

Question 2

• Ten basic types of equations are covered in this lesson. Five of these are included in the examples below. The other five
are x = y
2
, y = 1/x, y = [x], y = |x|, and y = x
3
.

Answer 2

• Ten basic types of equations are covered in this lesson. Five of these are included in the examples below. The other five
are x = y
2
, y = 1/x, y = [x], y = |x|, and y = x
3
.

Question 3

• y = 1/x
• y is equal to 1 divided by x
• called a rational function
• the graph of y = 1/x is two curves where one curve is in quadrant I and the other curve is in quadrant III
• five points contained on the graph of y = 1/x are (1, 1), (2, 1/2), (–1, –1), and (–2, –1/2)
• no points on the graph of y = 1/x have an x-coordinate equal to 0 or a y-coordinate equal to 0
• graphs of equations based on y = 1/x are two curves

Answer 3

• y = 1/x
• y is equal to 1 divided by x
• called a rational function
• the graph of y = 1/x is two curves where one curve is in quadrant I and the other curve is in quadrant III
• five points contained on the graph of y = 1/x are (1, 1), (2, 1/2), (–1, –1), and (–2, –1/2)
• no points on the graph of y = 1/x have an x-coordinate equal to 0 or a y-coordinate equal to 0
• graphs of equations based on y = 1/x are two curves

Question 4

• y = [x]
• y is equal to the greatest integer less than or equal to x
• called the greatest integer function
• the graph of y = [x] is a series of 1 unit long horizontal line segments where the left endpoint is a closed circle and the
right endpoint is an open circle
• five points contained on the graph of y = [x] are (0, 0), (1, 1), (–1, –1), (1/2, 0), and (–1/2, –1)
• the y-coordinate of each point on the graph of y = [x] is always an integer
• graphs of equations based on y = [x] are series of line segments

Answer 4

• y = [x]
• y is equal to the greatest integer less than or equal to x
• called the greatest integer function
• the graph of y = [x] is a series of 1 unit long horizontal line segments where the left endpoint is a closed circle and the
right endpoint is an open circle
• five points contained on the graph of y = [x] are (0, 0), (1, 1), (–1, –1), (1/2, 0), and (–1/2, –1)
• the y-coordinate of each point on the graph of y = [x] is always an integer
• graphs of equations based on y = [x] are series of line segments

Question 5

• y = |x|
• y is equal to the absolute value of x
• called the absolute value function
• the graph of y = |x| is V-shaped and opens upwards
• five points contained on the graph of y = |x| are (0, 0), (1, 1), (–1, 1), (2, 2) and (–2, 2)
• the y-coordinate of each point on the graph of y = |x| is always a positive number
• graphs of equations based on y = |x| are V-shaped graphs that open upwards or downwards

Answer 5

• y = |x|
• y is equal to the absolute value of x
• called the absolute value function
• the graph of y = |x| is V-shaped and opens upwards
• five points contained on the graph of y = |x| are (0, 0), (1, 1), (–1, 1), (2, 2) and (–2, 2)
• the y-coordinate of each point on the graph of y = |x| is always a positive number
• graphs of equations based on y = |x| are V-shaped graphs that open upwards or downwards

Question 6

• y = x
3
• y is equal to x cubed (to the third power)
• called the cubic function
• the graph of y = x
3
is a curve that extends infinitely in the negative direction (downwards) on the left end and infinitely
in the positive direction (upwards) on the right end
• five points contained on the graph of y = x
3
are (0, 0), (1, 1), (–1, –1), (2, 8) and (–2, –8)
• graphs of equations based on y = x
3
are curves that extend upwards on one end and downwards on the other end

Answer 6

• y = x
3
• y is equal to x cubed (to the third power)
• called the cubic function
• the graph of y = x
3
is a curve that extends infinitely in the negative direction (downwards) on the left end and infinitely
in the positive direction (upwards) on the right end
• five points contained on the graph of y = x
3
are (0, 0), (1, 1), (–1, –1), (2, 8) and (–2, –8)
• graphs of equations based on y = x
3
are curves that extend upwards on one end and downwards on the other end

Question 7

This graph is a straight line with a slope of 1.
y = x
If you try some points, you will find (1, 1),(–2, –2), (0, 0),
all on this line.
y =
Remember: x must be positive or 0 to be under the radical
sign. As a result, the graph does not exist to the left of 0 or
below the x-axis.
Points you will find on this line include (0, 0), (16, 4),
(25, 5) and (4, 2).

Answer 7

This graph is a straight line with a slope of 1.
y = x
If you try some points, you will find (1, 1),(–2, –2), (0, 0),
all on this line.
y =
Remember: x must be positive or 0 to be under the radical
sign. As a result, the graph does not exist to the left of 0 or
below the x-axis.
Points you will find on this line include (0, 0), (16, 4),
(25, 5) and (4, 2).

Question 8

y = 3sqrtx
When you try points with this expression, you will find
(1, 1), (–1, –1), (–8, –2), (27, 3)
On this graph both coordinates always have the same sign
and the x-values grow much faster than the y-values. Those
two characteristics place this graph in Quadrants I and III
following closely along the x-axis.

x = |y|
Try some points. You will find (0, 0), (2, 2) and (2, –2),
(5, 5) and (5, –5) among the points on this line.
There is no way for x to be negative on this graph.
y = x
2
Try some points and you will find (0, 0), (2, 4) and (–2, 4),
(5, 25) and (–5, 25).
There is no way for y to be negative and the y-values grow
much faster than the x-values.

Answer 8

y = 3sqrtx
When you try points with this expression, you will find
(1, 1), (–1, –1), (–8, –2), (27, 3)
On this graph both coordinates always have the same sign
and the x-values grow much faster than the y-values. Those
two characteristics place this graph in Quadrants I and III
following closely along the x-axis.

x = |y|
Try some points. You will find (0, 0), (2, 2) and (2, –2),
(5, 5) and (5, –5) among the points on this line.
There is no way for x to be negative on this graph.
y = x
2
Try some points and you will find (0, 0), (2, 4) and (–2, 4),
(5, 25) and (–5, 25).
There is no way for y to be negative and the y-values grow
much faster than the x-values.