Gradient Of A Curve: Finding Points On A Cubic Function

by Andrew McMorgan 56 views

What's up, math whizzes and curve conquerors! Today, we're diving deep into the fascinating world of calculus to tackle a super cool problem. We're going to find the exact coordinates of points on a specific graph where the gradient, which is basically the steepness or slope of the line touching the curve at that point, is exactly -3. The graph we're working with is a cubic function: y = 3x³ + 18x² + 24x + 5. This might sound a bit intimidating with all those powers of x, but trust me, guys, with a little calculus magic, we can break it down and find those elusive points. So, grab your calculators, your favorite study snacks, and let's get this mathematical party started!

Understanding Gradients and Derivatives

Alright, before we jump into solving for our specific points, let's do a quick refresher on what we mean by gradient in the context of a curve. Unlike a straight line, where the gradient is constant everywhere, the gradient of a curve changes from point to point. Think of it like driving up and down hills – sometimes you're going uphill steeply, sometimes downhill gently, and sometimes you're on a flat section. The gradient tells us exactly how steep the curve is at any given moment. In calculus, we find this instantaneous gradient using something called the derivative. The derivative of a function, often written as dy/dx or f'(x), gives us a new function that represents the gradient of the original function at every point. So, if our original function is y = f(x), its derivative dy/dx will be a function that outputs the gradient of y at any given x value. For our problem, we're given the function y = 3x³ + 18x² + 24x + 5, and we're told the gradient needs to be -3. This means we need to find the value(s) of x where dy/dx = -3.

Finding the Derivative of Our Cubic Function

Now, let's get our hands dirty and find the derivative of our cubic function, y = 3x³ + 18x² + 24x + 5. We'll use the power rule for differentiation, which is a fundamental tool in calculus. The power rule states that the derivative of axⁿ is anxⁿ⁻¹. Let's apply this to each term in our function:

  • For the term 3x³: Here, a = 3 and n = 3. So, the derivative is 3 * 3 * x^(3-1), which simplifies to 9x².
  • For the term 18x²: Here, a = 18 and n = 2. The derivative is 18 * 2 * x^(2-1), which simplifies to 36x.
  • For the term 24x: Remember that x is the same as x¹. So, here a = 24 and n = 1. The derivative is 24 * 1 * x^(1-1), which is 24 * x⁰. Since anything to the power of 0 is 1, this simplifies to just 24.
  • For the constant term 5: The derivative of any constant is always 0. So, the derivative of 5 is 0.

Putting it all together, the derivative of our function y = 3x³ + 18x² + 24x + 5 is:

dy/dx = 9x² + 36x + 24

This new function, 9x² + 36x + 24, is our gradient function. It tells us the gradient of the original curve at any value of x.

Setting the Gradient Equal to -3

We've found our gradient function, dy/dx = 9x² + 36x + 24. The problem states that we want to find the points where the gradient is exactly -3. So, all we need to do is set our gradient function equal to -3 and solve for x. This gives us the following equation:

9x² + 36x + 24 = -3

Our goal now is to solve this quadratic equation for x. First, let's move the -3 to the left side of the equation to set it to zero, which is the standard form for solving quadratic equations. We do this by adding 3 to both sides:

9x² + 36x + 24 + 3 = 0

9x² + 36x + 27 = 0

Lookin' good! We now have a quadratic equation. Notice that all the coefficients (9, 36, and 27) are divisible by 9. This is awesome because it means we can simplify the equation by dividing the entire equation by 9. This will make the numbers smaller and easier to work with:

(9x² + 36x + 27) / 9 = 0 / 9

x² + 4x + 3 = 0

This simplified quadratic equation is much easier to solve!

Solving the Quadratic Equation for x

We have the simplified quadratic equation x² + 4x + 3 = 0. There are a few ways to solve quadratic equations, like using the quadratic formula or factoring. Factoring is usually the quickest if it's possible, and this one looks like it's begging to be factored. We need to find two numbers that multiply to give us the constant term (3) and add up to give us the coefficient of the x term (4). Let's think...

  • Factors of 3 are (1, 3) and (-1, -3).
  • Which pair adds up to 4? That would be 1 and 3! (1 + 3 = 4).

So, we can factor our quadratic equation as:

(x + 1)(x + 3) = 0

For this equation to be true, one or both of the factors must equal zero. So, we set each factor equal to zero and solve for x:

  • Case 1: x + 1 = 0 Subtracting 1 from both sides gives us x = -1.

  • Case 2: x + 3 = 0 Subtracting 3 from both sides gives us x = -3.

Awesome! We've found two x-values where the gradient of the curve is -3: x = -1 and x = -3. These are the x-coordinates of the points we're looking for.

Finding the Corresponding y-coordinates

We've found the x-coordinates where the gradient is -3, but the question asks for the coordinates of the points, which means we need both the x and y values. To find the corresponding y-values, we need to plug these x-values back into the original function y = 3x³ + 18x² + 24x + 5.

Let's find the y-coordinate for x = -1:

y = 3(-1)³ + 18(-1)² + 24(-1) + 5

Remember the order of operations (PEMDAS/BODMAS) and that (-1)³ = -1 and (-1)² = 1.

y = 3(-1) + 18(1) + 24(-1) + 5

y = -3 + 18 - 24 + 5

y = 15 - 24 + 5

y = -9 + 5

y = -4

So, one of the points is (-1, -4).

Now, let's find the y-coordinate for x = -3:

y = 3(-3)³ + 18(-3)² + 24(-3) + 5

Let's calculate the powers first: (-3)³ = -27 and (-3)² = 9.

y = 3(-27) + 18(9) + 24(-3) + 5

y = -81 + 162 - 72 + 5

y = 81 - 72 + 5

y = 9 + 5

y = 14

So, the other point is (-3, 14).

Conclusion: The Coordinates We Found

And there you have it, folks! We've successfully navigated the twists and turns of a cubic function and found the exact locations where its gradient is -3. By understanding the relationship between a function and its derivative, we were able to set up and solve a quadratic equation. Our calculations show that the coordinates of the points on the graph of y = 3x³ + 18x² + 24x + 5 where the gradient is -3 are (-1, -4) and (-3, 14). Pretty neat, right? This process highlights the power of calculus in analyzing the behavior of functions. Whether you're studying for exams or just love flexing those math muscles, remember that breaking down complex problems step-by-step, understanding the core concepts, and practicing are key to mastering them. Keep exploring, keep calculating, and I'll catch you in the next math adventure!