sketch the region by the given curves y=1+sqrt(x),y=(3+x)/3
In google type:
functions graphs online
When you see list of results click on:
rechneronline.de/function-graphs/
When page be open in blue recatacangle type:
1+sqr(x)
in gray rectacangle type:
(3+x)/3
and click option Draw
This graph have one mistake.
Function 1+sqr(x) is not defined for x less of zero.
Don't draw blue line for x less of zero.
Other parts of graphs is correct.
To sketch the region defined by the curves y = 1 + sqrt(x) and y = (3 + x)/3, you can follow these steps:
1. Solve the equations to find the points of intersection.
Set the two equations equal to each other:
1 + sqrt(x) = (3 + x)/3
Multiply both sides by 3 to eliminate the fraction:
3 + 3sqrt(x) = 3 + x
Subtract 3 from both sides:
3sqrt(x) = x
Square both sides to eliminate the square root:
9x = x^2
Rearrange the equation:
x^2 - 9x = 0
Factor out x:
x(x - 9) = 0
Solve for x:
x = 0 and x = 9
Substitute these values into one of the original equations to find the corresponding y-coordinates:
For x = 0:
y = 1 + sqrt(0) = 1
For x = 9:
y = (3 + 9)/3 = 4
2. Plot the intersection points on a coordinate plane.
Mark the point (0, 1) and (9, 4) on the x-y plane.
3. Determine the behavior of each curve.
The curve y = 1 + sqrt(x) represents a square root function. It starts at (0, 1), increasing as x increases, and approaching infinity as x approaches infinity.
The curve y = (3 + x)/3 represents a linear function. It starts at (0, 1), increasing as x increases with a slope of 1/3.
4. Sketch the curves between the intersection points.
Draw the curve y = 1 + sqrt(x) starting from (0, 1) and curving upward.
Draw the curve y = (3 + x)/3 starting from (0, 1) and increasing with a slope of 1/3.
5. Shade the region between the two curves.
Shade the region bounded by the two curves, which is the region above the curve y = 1 + sqrt(x) and below the curve y = (3 + x)/3.
Now you have sketched the region defined by the given curves y = 1 + sqrt(x) and y = (3 + x)/3.