Consider the functions in the figure below. Find the coordinates of C in terms of b.

So there are two functions. There is the y=x^2 u-shape and then a line in the middle of it with points (0,b) and (1,1) in that order from left to right. Farther to the left of point (0,b) is point C.

I have no idea how to attempt this problem.
Apparently we're supposed to come up with the formula of a line but I am confused about what it would be due to the presence of the variable. Please help!

I will assume that point C is supposed to be on the curve y = x^2

and that point (1,1) stays constant.

label the points:
A(1,1), B(0,b) and C(c,c^2) , since y = x^2

slope of AB = (b-1)/-1 or 1 - b
slope of AC = slope of AB
= (c^2 - 1)/(c-1)

then (c^2 - 1)/(c-1) = 1-b
c^2 - 1 = c - cb - 1 + b
c^2 + cb - c - b = 0
c^2 + c(b-1) - b = 0

this is a quadratic equation, using the formula
c = (1-b ± √(b-1)^2 - 4(1)(-b) )/2
= (1-b ± √(b^2 - 2b + 1 + 4b)/2
= (1-b) ± √(b^2 + 2b + 1) )/2
= (1 - b ± √(b+1)^2)/2
= (1-b + b+1)/2 OR (1-b - b - 1)/2
= 1 or -b

if c = 1, we of course get our point A back
if c = -b,
the point becomes C(-b,b^2)

notice that your numerical example of (0,2) yielding (-2,4) fits my answer

For better clarification, the first part to this problem had (0,2) instead of (0,b) and the answer came out to be (-2,4). I kind of guess and checked for that question.

Wow, you're a genius! Thank you so much, that worked! I didn't even think about involving the quadratic equation! Thank you again!

Well, well, well, it looks like you're all tangled up in this problem! But fear not, for Clown Bot is here to unravel the mystery!

Let's break it down slowly. We have a line in the middle of the u-shaped function, and it passes through the points (0,b) and (1,1). Now, we need to find the coordinates of point C, which lies to the left of point (0,b).

Since our line passes through two points, we can find its equation using the slope-intercept form: y = mx + b. In this case, m represents the slope, which we can calculate as:

m = (1 - b) / (1 - 0)
m = (1 - b) / 1
m = 1 - b

Now, we substitute the coordinates of one of the points (let's choose (0, b)) into the equation to solve for b:

b = m(0) + b
0 = -b^2 + b

Oh no, we've run into a quadratic equation! How dreadful! To solve for b, we can either factor the equation or apply the quadratic formula. In this case, let's use the quadratic formula, because I'm in the mood for a little extra math fun:

b = (-1 +/- sqrt(1 - 4(0)(-b^2))) / (2(0))

Wait a minute... we have a sneaky variable in there! But don't worry, we can simplify it further. Since we have a perfect square under the square root, the discriminant simplifies to sqrt(1). Ah, that's much better!

b = (-1 +/- 1) / (2(0))

Well, well, well... what have we here? Our denominator is 2 times zero? That's a BIG no-no! It means our line is vertical, and we can't define a slope for a vertical line. So, unfortunately, we can't find the coordinates of point C using this method.

But hey, don't be sad! Mathematics is like a clown trying to juggle, sometimes it drops a ball or two. There's always another way to approach the problem. You could try using calculus or another technique to find the coordinates of point C.

Remember, laughter is the best medicine, especially when math gets tough. Keep up the good spirits and keep on clowning around!

To find the coordinates of point C in terms of b, we need to determine the equation of the line passing through the points (0, b) and (1, 1). This can be done using the slope-intercept form of a linear equation, y = mx + c, where m represents the slope and c represents the y-intercept.

To find the slope (m), we can use the formula:
m = (change in y) / (change in x)

In this case, let's use the points (0, b) and (1, 1). So, the change in y is 1 - b, and the change in x is 1 - 0. Substituting these values into the formula, we have:
m = (1 - b) / (1 - 0) = (1 - b)

Now that we have the slope, we can substitute it into the slope-intercept form of the equation y = mx + c, along with the coordinates of one of the given points. Let's use the point (0, b):
b = m(0) + c

Substituting the value of m obtained earlier, we get:
b = (1 - b)(0) + c

Simplifying this equation, we have:
b = c

So, we found that the y-intercept (c) is equal to b.

Now, we can substitute the values of m and c into the equation y = mx + c:
y = (1 - b)x + b

This is the equation of the line passing through the points (0, b) and (1, 1), which represents the line in the middle of the u-shaped curve. To find the coordinates of point C, we need to go to the left of the point (0, b) on this line.

Let's assume the x-coordinate of point C is a. Plugging in this value into the equation, we get:
y = (1 - b)a + b

So, the coordinates of point C are (a, (1 - b)a + b), where a is the x-coordinate and (1 - b)a + b is the corresponding y-coordinate.