Analyzing & Graphing F(x) = X^3 - 4x^2 - 9x + 36

by Andrew McMorgan 49 views

Alright, guys! Let's dive into the world of polynomials and take a closer look at the function f(x) = x³ - 4x² - 9x + 36. We're going to graph it and break down all its important features. Get ready to roll!

1. Initial Assessment

Before we even think about plotting points, let's get a feel for what we're dealing with. Our function f(x) = x³ - 4x² - 9x + 36 is a cubic polynomial because the highest power of x is 3. This tells us a few things right off the bat:

  • End Behavior: Cubic functions have different end behaviors depending on the sign of the leading coefficient (the number in front of the x³ term). In our case, the leading coefficient is 1 (positive). This means that as x goes to positive infinity, f(x) also goes to positive infinity. And as x goes to negative infinity, f(x) goes to negative infinity. In simpler terms, the graph will rise to the right and fall to the left.
  • Maximum Number of Turning Points: A cubic function can have at most two turning points (local maxima or minima). These are the points where the graph changes direction.
  • Maximum Number of x-intercepts: A cubic function can have up to three x-intercepts (also called roots or zeros). These are the points where the graph crosses the x-axis.

Knowing these characteristics helps us anticipate the general shape of the graph. We know it's going to be a curve that rises and falls, possibly with a couple of bumps along the way.

2. Finding the Intercepts

Intercepts are crucial points for graphing any function. They tell us where the graph crosses the x and y axes. Let's find them for our function f(x) = x³ - 4x² - 9x + 36.

a. y-intercept

The y-intercept is the point where the graph crosses the y-axis. This happens when x = 0. So, we plug in x = 0 into our function:

f(0) = (0)³ - 4(0)² - 9(0) + 36 = 36

Therefore, the y-intercept is at the point (0, 36). That's a pretty high intercept, which tells us our graph is shifted upwards quite a bit.

b. x-intercepts

The x-intercepts are the points where the graph crosses the x-axis. This happens when f(x) = 0. So, we need to solve the equation:

x³ - 4x² - 9x + 36 = 0

This is a cubic equation, and solving it directly can be tricky. However, we can try to factor it. Sometimes, you can guess a root by trying factors of the constant term (36 in this case). Let's try x = 3:

(3)³ - 4(3)² - 9(3) + 36 = 27 - 36 - 27 + 36 = 0

Great! x = 3 is a root. This means (x - 3) is a factor of the polynomial. Now we can use polynomial division (or synthetic division) to divide x³ - 4x² - 9x + 36 by (x - 3).

After performing the division, we get:

x³ - 4x² - 9x + 36 = (x - 3)(x² - x - 12)

Now we need to factor the quadratic x² - x - 12. This factors nicely into:

x² - x - 12 = (x - 4)(x + 3)

So, our completely factored equation is:

(x - 3)(x - 4)(x + 3) = 0

This gives us the x-intercepts:

  • x = 3 -> (3, 0)
  • x = 4 -> (4, 0)
  • x = -3 -> (-3, 0)

We have three x-intercepts, which is the maximum number possible for a cubic function. This tells us our graph will cross the x-axis three times.

3. Finding Critical Points (Local Maxima and Minima)

Critical points are the points where the function's derivative is either zero or undefined. These points are potential locations for local maxima and minima (the turning points of the graph). Since our function is a polynomial, its derivative is always defined. So, we just need to find where the derivative is zero.

a. Find the Derivative

First, we find the derivative of f(x) = x³ - 4x² - 9x + 36 using the power rule:

f'(x) = 3x² - 8x - 9

b. Set the Derivative to Zero and Solve

Now we set the derivative equal to zero and solve for x:

3x² - 8x - 9 = 0

This is a quadratic equation, and we can solve it using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Where a = 3, b = -8, and c = -9. Plugging these values into the quadratic formula, we get:

x = (8 ± √((-8)² - 4 * 3 * -9)) / (2 * 3) x = (8 ± √(64 + 108)) / 6 x = (8 ± √172) / 6 x = (8 ± 2√43) / 6 x = (4 ± √43) / 3

So, our critical points occur at:

  • x₁ = (4 + √43) / 3 ≈ 3.51
  • x₂ = (4 - √43) / 3 ≈ -0.84

c. Determine if Critical Points are Maxima or Minima

To determine whether these critical points are local maxima or minima, we can use the second derivative test. We find the second derivative of f(x):

f''(x) = 6x - 8

Now we evaluate the second derivative at our critical points:

  • f''((4 + √43) / 3) = 6 * ((4 + √43) / 3) - 8 = 2 * (4 + √43) - 8 = 2√43 > 0*. Since the second derivative is positive, this critical point is a local minimum.
  • f''((4 - √43) / 3) = 6 * ((4 - √43) / 3) - 8 = 2 * (4 - √43) - 8 = -2√43 < 0*. Since the second derivative is negative, this critical point is a local maximum.

d. Find the y-coordinates of the Critical Points

To complete the coordinates of our critical points, we plug our x values back into the original function f(x):

  • f(3.51) ≈ (3.51)³ - 4(3.51)² - 9(3.51) + 36 ≈ -6.03. So, the local minimum is approximately at (3.51, -6.03).
  • f(-0.84) ≈ (-0.84)³ - 4(-0.84)² - 9(-0.84) + 36 ≈ 39.12. So, the local maximum is approximately at (-0.84, 39.12).

4. Sketching the Graph

Now we have all the information we need to sketch a pretty accurate graph of f(x) = x³ - 4x² - 9x + 36.

  1. Plot the Intercepts: Plot the y-intercept (0, 36) and the x-intercepts (-3, 0), (3, 0), and (4, 0).
  2. Plot the Critical Points: Plot the local maximum (-0.84, 39.12) and the local minimum (3.51, -6.03).
  3. Consider End Behavior: Remember that the graph falls to the left and rises to the right.
  4. Connect the Points: Starting from the left, draw a curve that falls from negative infinity, passes through the x-intercept (-3, 0), rises to the local maximum, falls back down to the y-intercept (0, 36), continues to the x-intercept (3,0), dips to the local minimum, and then rises through the x-intercept (4,0) and continues to positive infinity.

5. Analyzing the Graph

Now that we've sketched the graph, let's summarize its key features:

  • Domain: All real numbers (-∞, ∞), since it's a polynomial.
  • Range: All real numbers (-∞, ∞), since it's a cubic function.
  • Intercepts: y-intercept at (0, 36), x-intercepts at (-3, 0), (3, 0), and (4, 0).
  • Local Maximum: Approximately at (-0.84, 39.12).
  • Local Minimum: Approximately at (3.51, -6.03).
  • Increasing Intervals: The function is increasing from (-∞, -0.84) and from (3.51, ∞).
  • Decreasing Interval: The function is decreasing from (-0.84, 3.51).
  • End Behavior: As x → -∞, f(x) → -∞. As x → ∞, f(x) → ∞.

Conclusion

So there you have it! We've successfully graphed and analyzed the cubic function f(x) = x³ - 4x² - 9x + 36. By finding the intercepts, critical points, and considering the end behavior, we were able to create a pretty accurate sketch and understand all the important features of this polynomial. Keep practicing, and you'll become a graphing pro in no time!