Question

describe and correct the error a student made when writing the quadratic function f(x)=2(x + 3)² - 4 in standard form.
the student did not
the function in standard form is f(x)=

f(x) = 2(x + 3)² - 4
f(x) = 2x² + 6x + 9 - 4
f(x) = 2x² + 6x + 5

distribute 2 to all the necessary terms.
compute 9 - 4 correctly.
compute (x + 3)² correctly.

Explanation:

Step1: Identify the error

The student started with \( f(x) = 2(x + 3)^2 - 4 \). To expand \( (x + 3)^2 \), we use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \). So \( (x + 3)^2 = x^2 + 6x + 9 \). Then we need to distribute the 2 to all terms inside the parentheses: \( 2(x^2 + 6x + 9)=2x^2 + 12x + 18 \), not \( 2x^2 + 6x + 9 \) as the student did. So the student did not distribute 2 to all the necessary terms.

Step2: Correct the expansion

First, expand \( (x + 3)^2 \): \( (x + 3)^2 = x^2 + 6x + 9 \).

Then distribute the 2: \( 2(x^2 + 6x + 9)=2x^2 + 12x + 18 \).

Now subtract 4: \( f(x)=2x^2 + 12x + 18 - 4 \).

Simplify the constant terms: \( 18 - 4 = 14 \).

So the correct standard form is \( f(x)=2x^2 + 12x + 14 \).

Answer:

The student did not distribute 2 to all the necessary terms. The correct function in standard form is \( f(x) = 2x^2 + 12x + 14 \).