Question

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}+8x + 0 ). write the standard form (vertex form) of the quadratic equation.
standard form: enter your answer
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Explanation:

Step1: Factor out the coefficient of $x^2$ from the first two terms

Given $y = 4x^{2}+8x + 0$. Factor out $4$ from $4x^{2}+8x$: $y=4(x^{2}+2x)+0$

Step2: Complete the square inside the parentheses

For the quadratic expression $x^{2}+2x$, using the formula $(a + b)^2=a^{2}+2ab + b^{2}$. Here $a = x$ and $2ab=2x$, so $b = 1$. Then $x^{2}+2x=(x + 1)^{2}-1$

Step3: Substitute back into the equation

Substitute $x^{2}+2x=(x + 1)^{2}-1$ into $y=4(x^{2}+2x)+0$: $y=4((x + 1)^{2}-1)+0$

Step4: Simplify the equation

Expand $4((x + 1)^{2}-1)$: $y=4(x + 1)^{2}-4$

Answer:

$y = 4(x + 1)^2-4$