Step-by-step explanation:
How do we set about finding the points in which two graphs y = f(x) and y = g(x) intersect?
We already know how to find where the graph of f(x) cuts the x−axis. That’s where y = 0. We calculate it by solving the equation f(x) = 0 .
When the graphs of y = f(x) and y = g(x) intersect , both graphs have exactly the same x and y values. So we can find the point or points of intersection by solving the equation f(x) = g(x). The solution of this equation will give us the x value(s) of the point(s) of intersection. We can then find the y value by putting the value for x that we have found into one of the original equations. That is by calculating either f(x) or g(x).
Example 1
Calculate the point of intersection of the two lines f(x) = 2x − 1 and g(x) = x + 1. First let’s look at a graph of the two functions. We can see the point of intersection is (2, 3).

We calculate the point of intersection by solving the equation f(x) = g(x). That is:
2x − 1 = x + 1
2x − x = 1 + 1
x = 2
The y coordinate can now be found by calculating f(2):
f(2) = 2×2 − 1 = 3
The point of intersection is (2, 3).
The example shows that we can find the point of intersection in two ways.
Either graphically, by drawing the two graphs in the same coordinate system, or algebraically by solving the equation such as the one in the above example.
Solving an equation graphically is easy with a graphical calculator or a computer program such as Excel.
Some equations cannot be solved algebraically but we can find solutions that are correct to as many significant figures as we want by using computers and calculators
Answer:
240 degrees
Step-by-step explanation:
Looking at the graph we can see that the first peak is at -30 degrees and the second peak is at 210 degrees.
210 - (-30) = 210 + 30 = 240
Answer:
I don't get the "number of 40" part, but here's my best shot!
3(40x)-5
Step-by-step explanation:
Given plane Π : f(x,y,z) = 4x+3y-z = -1
Need to find point P on Π that is closest to the origin O=(0,0,0).
Solution:
First step: check if O is on the plane Π : f(0,0,0)=0 ≠ -1 => O is not on Π
Next:
We know that the required point must lie on the normal vector <4,3,-1> passing through the origin, i.e.
P=(0,0,0)+k<4,3,-1> = (4k,3k,-k)
For P to lie on plane Π , it must satisfy
4(4k)+3(3k)-(-k)=-1
Solving for k
k=-1/26
=>
Point P is (4k,3k,-k) = (-4/26, -3/26, 1/26) = (-2/13, -3/26, 1/26)
because P is on the normal vector originating from the origin, and it satisfies the equation of plane Π
Answer: P(-2/13, -3/26, 1/26) is the point on Π closest to the origin.
