The graphs of functions f(x) and g(x) = f(x) + k are shown below:

Mathematics

2 answers:

allochka39001 [22]3 years ago

8 0

The correct answer for k would be 2

Nataly [62]3 years ago

7 0

Here's the info for f(x): We are going to find the slope of the line and then write the equation for the line using one of the given points. The coordinate points we are given are (0, 0) and (2, 4). Using the slope formula:
$m= \frac{y_{2} - y_{1} }{ x_{2}- x_{1} }$
gives us a slope equation of:
$m= \frac{4-0}{2-0}$ and the slope is 2. Using the point (0, 0) to write the equation of the line for f(x) looks like this in the slope-intercept form of the equation:
$y- y_{1} =m(x- x_{1})$ where m is the sloppe of 2 that we found and $y_{1}$ and $x_{1}$ are the coordinates of one of the points. It doesn't matter which one you choose; you will get the same answer whether you use (0, 0) or (2, 4): y-0=2(x-0) Distributing that 2 into the parenthesis and simplifying gives you the equation of y = 2x, or in our function notation, f(x) = 2x. Since f(x) is the first part of g(x), so far for g(x) we have that g(x) = 2x + k. Now we will do the same thing for g(x) that we did for f(x) as far as writing its equation down; we don't need to find the slope cuz the slope of g(x) is the function f(x). The equation for g(x), using the point (0, 2) (again, you could have used either point; I just picked (0, 2) cuz the other one has a decimal in it!): y - 2 = 2(x - 0). Distributing that 2 into the parenthesis gives you this: y - 2 = 2x - 0; y = 2x + 2. So 2 is your k value!

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emmasim [6.3K]

a. The general equation for a circle centered at $(a,b)$ with radius $r$ is

$(x-a)^2+(y-b)^2=r^2$

The described circle has equation

$(x+3)^2+(y+2)^2=r^2$

We know the circle passes through the origin. This means that the equation above holds for $x=0$ and $y=0$ . The distance between any point on the circle and its center is the radius, so we can use this fact to determine $r$ :

$(0+3)^2+(0+2)^2=r^2\implies 9+4=13=r^2\implies r=\sqrt{13}$

So the circle's equation is

$(x+3)^2+(y+2)^2=(\sqrt{13})^2=13$

b. If the distance between point B and the center is less than $\sqrt{13}$ , then B lies inside the circle. If the distance is greater than $\sqrt{13}$ , it falls outside the circle. Otherwise, if the distance is exactly $\sqrt{13}$ , then B lies on the circle.