Question

test 3 b
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current learning objective: understanding how the graphs of parabolas are related to their quadratic functions
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a quadratic function has the equation ( y = 4x^{2}+16x + 14 ). write the standard form (vertex form) of the quadratic equation.
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Explanation:

Step1: Factor out the coefficient of \(x^{2}\) from the first two terms

Given \(y = 4x^{2}+16x + 14\), factor out \(4\) from \(4x^{2}+16x\):
\(y=4(x^{2}+4x)+14\)

Step2: Complete the square inside the parentheses

For the quadratic expression \(x^{2}+4x\), use the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\). Here \(a=x\) and \(2ab = 4x\), so \(b = 2\). Then \(x^{2}+4x=(x + 2)^{2}-4\)
Substitute back into the equation: \(y=4((x + 2)^{2}-4)+14\)

Step3: Expand and simplify

Expand \(4((x + 2)^{2}-4)\): \(y=4(x + 2)^{2}-16+14\)
Simplify: \(y=4(x + 2)^{2}-2\)

Answer:

\(y = 4(x + 2)^{2}-2\)