Score: \( 1 / 10 \) Penalty: 0.25 off Question Re-write the quadratic function below in Standard Form Show Examples

To rewrite a quadratic function in standard form, we typically express it in the form: \[ f(x) = a(x - h)^2 + k \] where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola. ### Example 1: Let's take the quadratic function: \[ f(x) = x^2 + 4x + 3 \] **Step 1: Identify coefficients.** - Here, \(a = 1\), \(b = 4\), and \(c = 3\). **Step 2: Complete the square.** 1. Take half of the coefficient of \(x\) (which is \(4\)), square it, and add/subtract it inside the function: - Half of \(4\) is \(2\), and squaring it gives \(4\). 2. Rewrite the function: \[ f(x) = (x^2 + 4x + 4) - 4 + 3 \] \[ f(x) = (x + 2)^2 - 1 \] **Step 3: Write in standard form.** - The standard form is: \[ f(x) = 1(x + 2)^2 - 1 \] ### Example 2: Consider another quadratic function: \[ f(x) = 2x^2 - 8x + 6 \] **Step 1: Identify coefficients.** - Here, \(a = 2\), \(b = -8\), and \(c = 6\). **Step 2: Complete the square.** 1. Take half of \(-8\) (which is \(-4\)), square it, and add/subtract it: - Half of \(-8\) is \(-4\), and squaring it gives \(16\). 2. Rewrite the function: \[ f(x) = 2(x^2 - 4x + 16) - 16 + 6 \] \[ f(x) = 2(x - 4)^2 - 10 \] **Step 3: Write in standard form.** - The standard form is: \[ f(x) = 2(x - 4)^2 - 10 \] ### Summary To convert a quadratic function to standard form: 1. Identify coefficients \(a\), \(b\), and \(c\). 2. Complete the square. 3. Rewrite the function in the form \(f(x) = a(x - h)^2 + k\). If you have a specific quadratic function you would like to convert to standard form, please provide it, and I can assist you further!