Final answer:
The function that translates each point on the graph of y=mx+b so that the slope of the segment from the original point to the translated point is 43 is y=mx+(b+43).
Step-by-step explanation:
To understand this concept, we first need to understand what a slope is. In simple terms, slope is the measure of how steep a line is. It is defined as the change in the y-coordinate divided by the change in the x-coordinate. In other words, it represents the rate at which the line is changing.
Now, let's look at the original equation y=mx+b. Here, m represents the slope of the line and b represents the y-intercept, which is the point where the line crosses the y-axis. So, if we want the slope of the segment from the original point to the translated point to be 43, we need to find a way to change the value of m.
To do this, we can add a constant value to the y-intercept b. This will change the position of the line on the y-axis, but it will not affect the slope. So, if we add a value of 43 to b, the new equation will be y=mx+(b+43). This means that the slope remains unchanged, but the line has shifted upwards by 43 units.
But why does this work? Let's take a closer look. The slope of the line is given by the formula m=y2-y1/x2-x1, where (x1,y1) and (x2,y2) are any two points on the line. Now, when we add 43 to b, the y-intercept of the translated line becomes b+43. So, the y-coordinate of the translated point will be y1+43. This means that the difference between the y-coordinates of the original point and the translated point will be y1+43-y1=43. Similarly, the difference between the x-coordinates of the two points will remain the same. Hence, the slope of the segment will be 43/1=43, as desired.
Another way to think about it is that the constant added to the y-intercept shifts the entire line upwards, but it does not change the steepness of the line. So, the slope remains unchanged.
In conclusion, the function y=mx+(b+43) translates each point on the graph of y=mx+b so that the slope of the segment from the original point to the translated point is 43. This is achieved by adding 43 to the y-intercept, which does not affect the slope but shifts the line upwards by 43 units.