Question:** An angel investor is evaluating a startup's growth model, represented by the cubic polynomial \( f(t) = t^3 + pt^2 + qt + r \), where \( t \) is time in years. If \( f(0) = 2 \), \( f(1) = 0 \), and \( f(2) = -2 \), find the coefficients \( p \), \( q \), and \( r \).

["Title: How to Determine Growth Parameters Using Given Data: Solving for Coefficients in a Cubic Growth Model", "---", "When evaluating a startup’s growth trajectory, angel investors often rely on mathematical models to assess potential and sustainability. One powerful tool is a cubic polynomial model:\n[\nf(t) = t^3 + pt^2 + qt + r\n]\nwhere ( t ) represents time in years, and ( p ), ( q ), and ( r ) are key parameters defining the startup’s growth pattern. In this article, we explore how to determine these coefficients using real-world data points: ( f(0) = 2 ), ( f(1) = 0 ), and ( f(2) = -2 ). Understanding these values helps investors predict future performance, identify risks, and validate the startup’s business model.", "### The Role of Initial Conditions in Model Calibration", "The polynomial form is given with a leading coefficient of 1 (indicating pure cubic growth over linear scaling), so:\n[\nf(t) = t^3 + pt^2 + qt + r\n]\nWe are provided three concrete values:", "- At ( t = 0 ): ( f(0) = 2 )\n- At ( t = 1 ): ( f(1) = 0 )\n- At ( t = 2 ): ( f(2) = -2 )", "Using these, we can set up a system of equations to solve for the unknowns ( p ), ( q ), and ( r ).", "---", "### Step 1: Use ( f(0) = 2 ) to find ( r )", "Substitute ( t = 0 ):\n[\nf(0) = 0^3 + p(0)^2 + q(0) + r = r\n]\nGiven ( f(0) = 2 ), we immediately get:\n[\nr = 2\n]", "---", "### Step 2: Use ( f(1) = 0 ) to form the first equation", "Substitute ( t = 1 ):\n[\nf(1) = 1^3 + p(1)^2 + q(1) + r = 1 + p + q + r\n]\nGiven ( f(1) = 0 ) and ( r = 2 ), we have:\n[\n1 + p + q + 2 = 0\n]\nSimplify:\n[\np + q = -3 \quad \ ext{(Equation 1)}\n]", "---", "### Step 3: Use ( f(2) = -2 ) to form the second equation", "Substitute ( t = 2 ):\n[\nf(2) = 2^3 + p(2)^2 + q(2) + r = 8 + 4p + 2q + r\n]\nGiven ( f(2) = -2 ) and ( r = 2 ):\n[\n8 + 4p + 2q + 2 = -2\n]\nSimplify:\n[\n10 + 4p + 2q = -2\n]\n[\n4p + 2q = -12\n]\nDivide entire equation by 2:\n[\n2p + q = -6 \quad \ ext{(Equation 2)}\n]", "---", "### Step 4: Solve the system of equations", "We now solve:\n[\n\begin{cases}\np + q = -3 & \ ext{(1)} \\n2p + q = -6 & \ ext{(2)}\n\end{cases}\n]", "Subtract Equation 1 from Equation 2:\n[\n(2p + q) - (p + q) = -6 - (-3) \Rightarrow p = -3\n]", "Substitute ( p = -3 ) into Equation 1:\n[\n-3 + q = -3 \Rightarrow q = 0\n]", "---", "### Final Coefficients", "- ( p = -3 )\n- ( q = 0 )\n- ( r = 2 )", "Thus, the startup’s growth model is:\n[\nf(t) = t^3 - 3t^2 + 2\n]", "---", "### Why This Matters for Angel Investors", "By rooting the growth model in empirical data, investors can verify whether a startup’s projected trajectory aligns with reality. The cubic curve ( f(t) = t^3 - 3t^2 + 2 ) suggests accelerating deceleration—common in scaling startups where growth slows after early momentum. Recognizing such patterns helps assess sustainability, potential market capture, and the need for strategic pivots.", "Understanding how to extract model parameters from key data points empowers investors to make informed decisions, balancing innovation with measurable evidence.", "---", "Keywords: angel investor, startup growth model, cubic polynomial, cubic regression, angel investment due diligence, f(0)=2, f(1)=0, f(2)=-2, investor modeling, polynomial growth coefficients."]