Question

(see example 4.)

1. $f(x) = 4x - 9 - x^3$

28.

Explanation:

Step1: Analyze the function \( f(x) = -x^3 + 4x - 9 \)

The function is a cubic function. The leading term is \( -x^3 \), so as \( x \to +\infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to +\infty \) (since the leading coefficient is negative and the degree is odd).

Step2: Find some key points

• When \( x = 0 \), \( f(0) = -0 + 0 - 9 = -9 \).
• When \( x = 1 \), \( f(1) = -1 + 4 - 9 = -6 \).
• When \( x = 2 \), \( f(2) = -8 + 8 - 9 = -9 \).
• When \( x = 3 \), \( f(3) = -27 + 12 - 9 = -24 \).
• When \( x = -1 \), \( f(-1) = 1 - 4 - 9 = -12 \).
• When \( x = -2 \), \( f(-2) = 8 - 8 - 9 = -9 \).

Step3: Sketch the graph (qualitative)

Using the end - behavior and the key points, we can sketch the graph. The cubic function will have a general "S - shaped" curve (but since the leading coefficient is negative, it will be decreasing for large \( |x| \) with the appropriate turning points. The derivative \( f^\prime(x)=-3x^{2}+4 \), setting \( f^\prime(x) = 0 \), we get \( x=\pm\sqrt{\frac{4}{3}}\approx\pm1.15 \). At \( x = \sqrt{\frac{4}{3}} \), \( f(x)=-(\frac{4}{3})^{\frac{3}{2}}+4\sqrt{\frac{4}{3}}-9\approx - 1.54 + 2.31-9=-8.23 \). At \( x=-\sqrt{\frac{4}{3}} \), \( f(x)=(\frac{4}{3})^{\frac{3}{2}}-4\sqrt{\frac{4}{3}}-9\approx1.54 - 2.31 - 9=-9.77 \).

To plot on the given grid (assuming the grid has appropriate x - y axes with integer coordinates for simplicity), we can mark the points we calculated: (0, - 9), (1, - 6), (2, - 9), (3, - 24), (- 1, - 12), (- 2, - 9) and then draw the curve passing through these points, considering the end - behavior and the critical points from the derivative.

(Note: Since the problem says "See Example 4" and we assume Example 4 is about graphing cubic functions, the main idea is to use the properties of cubic functions - end - behavior, key points, and critical points from the derivative - to sketch the graph of \( f(x)=-x^{3}+4x - 9 \))

Answer:

The graph of \( f(x)=4x - 9 - x^{3}=-x^{3}+4x - 9 \) is a cubic curve with the properties and key points as analyzed above, and can be sketched on the given grid using the steps of analyzing end - behavior, finding key points, and using the derivative for critical points.