To identify a function that matches the given graph, we analyze its key characteristics:

Observations:

1. Degree of the polynomial: The graph has three x-intercepts and two turning points, suggesting that this is a cubic function (degree 3). A cubic polynomial can be written as: $f(x) = a(x - r_1)(x - r_2)(x - r_3),$ where $r_1, r_2, r_3$ are the roots of the equation.

2. X-intercepts (roots): From the graph, the approximate roots (where $f(x) = 0$) are: $x = 0, \, x = 2, \, x = 4.$

3. Behavior at the roots:

   • The graph touches the x-axis at $x = 0$ (multiplicity 2 root).
   • It crosses the x-axis at $x = 4$. This suggests that $x = 0$ is a repeated root (multiplicity 2), and $x = 4$ is a simple root.

4. Leading coefficient and end behavior: The graph rises sharply as $x \to \infty$ and falls sharply as $x \to -\infty$, meaning the leading coefficient $a > 0$.

General Form of the Function:

Combining these observations, the function can be written as: $f(x) = a(x)^2(x - 4),$ where $a$ is a constant that determines the steepness of the curve.

Solving for $a$:

To find $a$, we use another known point from the graph. For example, at $x = 2$, the graph passes through approximately $y = -8$. Substituting $x = 2$ and $y = -8$ into the function: $f(2) = a(2)^2(2 - 4) = -8.$ Simplify: $a(4)(-2) = -8.$ $-8a = -8 \implies a = 1.$

Final Function:

The function that matches the graph is: $f(x) = x^2(x - 4).$

Verification:

• At $x = 0$, $f(0) = 0$ (root at $x = 0$, multiplicity 2).
• At $x = 4$, $f(4) = 0$ (simple root at $x = 4$).
• At $x = 2$, $f(2) = -8$, which matches the graph.

This confirms the function is correct.

Do you want more details or explanations about any step?

Related Questions:

1. What is the general form of a polynomial with multiple roots?
2. How do you determine the multiplicity of roots from a graph?
3. How can you identify the degree of a polynomial based on its turning points?
4. How do you solve for coefficients of polynomials using known points?
5. What is the significance of the leading coefficient in polynomial graphs?

Tip: A root's multiplicity affects whether the graph touches or crosses the x-axis. Multiplicity 2 causes the graph to "bounce" at the root.