Problem: Find the equation of the normal to the curve f(x) = x2 + 4x - 2 at the point where x = -3. Find the coordinates of the other point where this point intersects the curve again.
This problem was taken from the Pearson Textbook, on page 619. (Lesson 13.2, exercise 6). Give this problem a go, if you want, then look below to see how I solved this question.
Solution:
The first thing to do is actually figure out what point we're finding the normal of. And we can't make the normal without the tangent. A normal is the line perpendicular to the tangent of a function. The tangent of a graph is the instantaneous rate of change at a specific point.
f(x) = x2 + 4x - 2
f(-3) = (-3)2 + 4(-3) -2
f(-3) = 9 - 12 - 2
f(-3) = -5
Now we have the point (-3,-5). Next, in order to find the tangent, we need the instantaneous rate of change. Using the slope formula learned back in middle school won't work, because we'll end up with 0/0, which is indeterminate form. What we need is to find the derivative at that point. Now if you don't know what derivatives are, they basically tell us the slope of the tangent at any given point. And if you didn't see my cubic exploration, here's a quick run-down of the only rule you'll need to know for this problem:
f(x) = axn
f'(x) = anxn-1
In this case, f'(x) = 2x + 4 . Now, we plug in our x value to find the instantaneous rate of change.
f(-3) = 2(-3) + 4
f(-3) = -6 + 4
f(-3) = -2
After that, we can make the equation of the tangent. I will be making this in point-slope form, because it is easy to make, and doesn't really need to be simplified.
y - y1 = m(x - x1)
y + 5 = -2(x + 3)
The equation of the normal will be the same, but the slope (m) will be reciprocated and turned positive. Because this will lead to the answer, I will simplify this one.
y + 5 = 0.5(x + 3)
y = 0.5x +1.5 - 5
y = 0.5x - 3.5
Now, put this equation equal to the original, and get everything to one side.
0.5x - 3.5 = x2 + 4x - 2
0 = x2 + 3.5x + 1.5
We already know one of the zeroes will be at x = -3 because that data was given to us, so we can divide this function by -3. To divide polynomials, we will use synthetic division because it's easiest. I highly recommend you check out that cubics exploration I keep mentioning, because it explains what I'm talking about.
Well when we divide x2 + 3.5x + 1.5 by -3, we get (x + 0.5). Therefore, we can conclude the other point in which the normal interests is at x = -0.5. Now, all we have to do is plug that into our original equation to get the other point of intersection.
f(x) = x2 + 4x - 2
f(-0.5) = (-0.5)2 + 4(-0.5) - 2
f(-0.5) = 0.25 - 2 - 2
f(-0.5) = -3.75
Therefore, we can conclude the second point is located at (-0.5, -3.75). We can check this graphically as well. Below is a graph from the Desmos Graphing Calculator, proving this answer is correct.
Of course, we could have used the graphing calculator from the very start, however it's always good to know how to solve this type of question without a calculator in case you don't have access to one.
This problem was taken from the Pearson Textbook, on page 619. (Lesson 13.2, exercise 6). Give this problem a go, if you want, then look below to see how I solved this question.
Solution:
The first thing to do is actually figure out what point we're finding the normal of. And we can't make the normal without the tangent. A normal is the line perpendicular to the tangent of a function. The tangent of a graph is the instantaneous rate of change at a specific point.
f(x) = x2 + 4x - 2
f(-3) = (-3)2 + 4(-3) -2
f(-3) = 9 - 12 - 2
f(-3) = -5
Now we have the point (-3,-5). Next, in order to find the tangent, we need the instantaneous rate of change. Using the slope formula learned back in middle school won't work, because we'll end up with 0/0, which is indeterminate form. What we need is to find the derivative at that point. Now if you don't know what derivatives are, they basically tell us the slope of the tangent at any given point. And if you didn't see my cubic exploration, here's a quick run-down of the only rule you'll need to know for this problem:
f(x) = axn
f'(x) = anxn-1
In this case, f'(x) = 2x + 4 . Now, we plug in our x value to find the instantaneous rate of change.
f(-3) = 2(-3) + 4
f(-3) = -6 + 4
f(-3) = -2
After that, we can make the equation of the tangent. I will be making this in point-slope form, because it is easy to make, and doesn't really need to be simplified.
y - y1 = m(x - x1)
y + 5 = -2(x + 3)
The equation of the normal will be the same, but the slope (m) will be reciprocated and turned positive. Because this will lead to the answer, I will simplify this one.
y + 5 = 0.5(x + 3)
y = 0.5x +1.5 - 5
y = 0.5x - 3.5
Now, put this equation equal to the original, and get everything to one side.
0.5x - 3.5 = x2 + 4x - 2
0 = x2 + 3.5x + 1.5
We already know one of the zeroes will be at x = -3 because that data was given to us, so we can divide this function by -3. To divide polynomials, we will use synthetic division because it's easiest. I highly recommend you check out that cubics exploration I keep mentioning, because it explains what I'm talking about.
Well when we divide x2 + 3.5x + 1.5 by -3, we get (x + 0.5). Therefore, we can conclude the other point in which the normal interests is at x = -0.5. Now, all we have to do is plug that into our original equation to get the other point of intersection.
f(x) = x2 + 4x - 2
f(-0.5) = (-0.5)2 + 4(-0.5) - 2
f(-0.5) = 0.25 - 2 - 2
f(-0.5) = -3.75
Therefore, we can conclude the second point is located at (-0.5, -3.75). We can check this graphically as well. Below is a graph from the Desmos Graphing Calculator, proving this answer is correct.
Of course, we could have used the graphing calculator from the very start, however it's always good to know how to solve this type of question without a calculator in case you don't have access to one.
