Determine the x-intercept and the y-intercept of the following lines:

Question

Determine the x-intercept and the y-intercept of the following lines:

 

a) 2x + 4y = 3

 

b) y = √2 x - 2/3

 

 

c) x = -2

 

d) y = 0

 

e) x = 2y

Answer 1

Recall that the x-intercept is the value of x where f(x)=0, and the y-intercept is the value of y where x=0.

 

The x-intercept and the y-intercept of the following lines are as follows.

a) 2x + 4y = 3

 

x-intercept:

 

2x+4(0)=3

2x=3

x=3/2

 

y-intercept:

 

2(0)+4y=3

4y=3

y=3/4

b) y = √2 x - 2/3

x-intercept:

 

0=√2x-2/3

√2x=2/3

x=(2/3)(1/√2)=2/(3√2)=(2√2)/6=√2/3

 

y-intercept:

 

y=-2/3 (by slope intercept form rule)

c) x = -2

 

x-intercept:

 

x=-2

 

y-intercept:

 

y=undefined

 

d) y = 0

 

x-intercept:

 

x=the set of real numbers

 

y-intercept:

 

y=0

e) x = 2y

 

x-intercept:

 

x=0

 

y-intercept:

 

y=0

Answer 2

A) 2x + 4y = 3:  What is the value of x when y is zero?  2x + 4(0) = 3.  This value will be the x intercept.  It won't be on the line above the x.axis where values of y are positive, won't be on the line below the x.axis where y is negative, but it will be on the line where the line has a value of zero for y, i.e. it will also be on the x.axis.  Solve 2x = 3 for x and this will be the value of x when y is zero.

 

Similarly, what is the value of y on this line when x is zero?  2(0) + 4 y = 3.  Solve 4y = 3 for y to get the y.intercept.

B) y = √2 x - 2/3:  Here the y intercept is easy because it is in slope-intercept form of y = mx + b.  b is the y.intercept.  Careful.  Is b = +2/3 or -2/3.

Now the x intercept is more difficult.  Solving for x yields  x = (y + 2/3)/√¯₂   and when y = 0, x is (√¯2) / 3.

c) x = -2: sketch the x-y coordinate plane.  Lay your pencil vertically along the y.axis. While maintaining the attitude of your pencil, displace it 2 units to the left of the y.axis.  What axis is intercepted - the x or the y?  What is the x intercept.  Does it intercept the y axis?

 

y is not defined here.  There is nothing to tell you what y is.  y is from -∞ to ∞.  So y can be a lot of things.

y can even be equal to y = (x/0.00000000000000000000000000000000000000000000001) - 2, but...

 

but y CAN NOT be y = x/0 -2.  Thou shalt not divide by zero.

 

d) y = 0:  mark all the points on the x-y cartesian coordinate plane where y = 0.  Ok, you can stop....Hey!  Stop!  Oops....  Well, if you ever stop, and read this, what did all of those points form?

e) x = 2y:   It won't hurt to solve for y, or simply divide by 2 on both sides:

 

y = x/2 or y = (1/2 ) x, or y = (1/2) x + 0.

 

So now it is in slope intercept form and the y.intercept is 0 and back to x = 2y, x = 0 when y is 0, so the x.intercept is 0.