To **find the cubic polynomial function from its graph**, you generally need to **identify key points and features** of the graph. Here’s a step-by-step guide: --- ### 1. **Understand the General Form** A cubic polynomial has the form: \[ f(x) = ax^3 + bx^2 + cx + d \] where \(a, b, c, d\) are constants. --- ### 2. **Identify Key Points on the Graph** - **x-intercepts (roots):** Points where the graph crosses the x-axis (\(f(x) = 0\)). - **y-intercept:** Point where the graph crosses the y-axis (\(x = 0\)). - **Turning points (local maxima/minima):** Where the graph changes direction. - **Inflection point:** Where the graph changes concavity (for cubics, always one inflection). --- ### 3. **Collect Data Points** Pick **four distinct points** on the graph. Ideally: - The y-intercept (easy to read) - Three x-intercepts, **or** any three other points (especially if not all roots are visible or rational) Let’s call your points: \((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)\). --- ### 4. **Set Up the System of Equations** Plug each point into the cubic form: \[ \begin{cases} y_1 = a{x_1}^3 + b{x_1}^2 + c{x_1} + d \\ y_2 = a{x_2}^3 + b{x_2}^2 + c{x_2} + d \\ y_3 = a{x_3}^3 + b{x_3}^2 + c{x_3} + d \\ y_4 = a{x_4}^3 + b{x_4}^2 + c{x_4} + d \\ \end{cases} \] Solve this system for \(a, b, c, d\). --- ### 5. **(Optional) Use Factored Form if Roots are Known** If you can see all three roots (\(r_1, r_2, r_3\)), the cubic can be written as: \[ f(x) = a(x - r_1)(x - r_2)(x - r_3) \] Find \(a\) by plugging in the y-intercept (\(x = 0\)). --- ### 6. **Double-Check with Another Point** After finding the coefficients, check if another visible point fits the equation for accuracy. --- ## **Example** Suppose from the graph you read these points: - x-intercepts: \((-2, 0)\), \((1, 0)\), \((3, 0)\) - y-intercept: \((0, 6)\) Write in factored form: \[ f(x) = a(x + 2)(x - 1)(x - 3) \] Plug in y-intercept: \[ 6 = a(0 + 2)(0 - 1)(0 - 3) = a(2)(-1)(-3) = a(6) \] \(\implies a = 1\) So, \(f(x) = (x + 2)(x - 1)(x - 3)\) --- ## **Summary Steps** 1. Identify points (especially intercepts) on the graph. 2. Plug at least four points into \(ax^3 + bx^2 + cx + d\). 3. Solve for coefficients. 4. Check your answer with other points. --- **Note:** If you only have the graph, your answer will be as accurate as your reading of the points! For higher precision, pick points where the graph clearly passes through integer values.